A Study of LN-144 or LN-145 in People With Advanced Uveal Melanoma, Undifferentiated Pleomorphic Sarcoma, or Dedifferentiated Liposarcoma

Inclusion Criteria: Cohort 1: Must have a confirmed diagnosis of metastatic Uveal Melanoma. * Patients will be eligible regardless of the number of prior systemic therapies received. * Cohort 2: Must have a confirmed diagnosis of unresectable or metastatic undifferentiated pleomorphic sarcoma (UPS) or dedifferentiated liposarcoma (DDLPS) that is refractory to at least 1 prior line of systemic therapy * Unresectable disease will be defined by an expert sarcoma surgical onocologist as either (a) low liklihood of obtaining an R0 resection or (b) unacceptable morbidity from a surgical procedure * Prior systemic therapy in the neoadjuvant or adjuvant setting will count has prior systemic therapy * Patients who refuse standard of care chemotherapy will be eligible * One (1) lesion at least 1.5cm in size (solitary or aggregate) available for TIL harvesting that has not undergone prior embolization or RT in prior 3 months unless subsequent growth is demonstrated (at least 0.5cm). * Patients must be ≥ 18 years of age at the time of consent. * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Patients must have an estimated life expectancy of ≥ 6 months in the opinion of the Investigator. * Patients must have the following hematologic parameters: * Absolute neutrophil count (ANC) ≥ 1000/mm3 * Hemoglobin (Hb) ≥ 9.0 g/dL * Platelet ≥ 100,000/mm\^3 Note: Transfusions or growth factors are not allowed 28 days prior to signing the ICF and continuing through the Screening Period * Patients must have adequate organ function: * Serum alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤ 3 times the upper limit of normal (≤ 3 × ULN); patients with liver metastasis ≤ 5 × ULN * Estimated creatinine clearance (eCrCl) ≥ 40 mL/min using the Cockcroft-Gault formula at Screening * Total bilirubin ≤ 2 mg/dL * Patients with Gilbert's syndrome must have a total bilirubin ≤ 3 mg/dL * Patients must be seronegative for the following: * Human immunodeficiency virus (HIV)-1 or HIV-2 antibodies * Hepatitis B antigen (HBsAg), hepatitis B core antibody (anti- HBc), or hepatitis C antibody (HCV Ab). Patients with acute or chronic hepatitis infections may be enrolled if the viral load by polymerase chain reaction (PCR) is undetectable with/without active treatment. * Syphilis (Rapid Plasma Reagin \[RPR\] test or venereal disease research laboratory \[VDRL\] test) * Cytomegalovirus (CMV) IgM antibody titer or PCR assay; and Epstein-Barr virus (EBV) IgM or PCR assay indicating active infection * Herpes simplex virus (HSV)-1 and HSV-2 IgM serology or PCR assay * Patients who are HSV immunoglobulin M (IgM) or PCR assay positive will need to receive appropriate treatment and become IgM or PCR assay negative prior to starting the NMA-LD pre-conditioning regimen * Anyone with prior COVID-19 infection must be asymptomatic for \>30 days prior to NMA-LD. * Patients must have a washout period from prior anticancer therapy(ies) of a minimum duration, as detailed below prior to the first study treatment (ie, start of NMA-LD): * Targeted therapy: prior targeted therapy with an EGFR, MEK, BRAF, ALK, ROS1, or other-targeted agents (eg, erlotinib, afatinib, dacomitinib, osimertinib, crizotinib, ceritinib, lorlatinib) is allowed provided the washout is a minimum of 14 days or 5 half-lives (whichever is longer) prior to the start of treatment * Chemotherapy: minimum of 21 days prior to the start of treatment * Immunotherapy: checkpoint-targeted therapy with an anti PD-1/anti PD-L1, other monoclonal antibodies, or vaccines are allowed, provided the washout is a minimum of 21 days prior to the start of study treatment * Palliative radiation therapy is permitted so long as it does not involve lesions being selected for TIL, or as target or non-target lesions. Washout is not required if all related toxicities have resolved to ≤ Grade 1 as per CTCAE v 5.0. * Patients must have recovered from all prior anti-cancer therapy-related adverse events (AEs) to ≤ Grade 1 (per Common Terminology Criteria for Adverse Events \[CTCAE\] v 5.0), except for alopecia or vitiligo, prior to enrollment. * Patients with documented ≥ Grade 2 diarrhea or colitis as a result of previous treatment with immune checkpoint inhibitor(s) must have been asymptomatic for at least 6 months and/or had a normal colonoscopy post-immune checkpoint inhibitortreatment, by visual assessment, prior to tumor resection. * Patients with immunotherapy-related endocrinopathies (e.g. hypothyroidism) stable for at least 6 weeks and controlled with hormonal replacement are allowed. * Previous surgical procedure(s) is/are permitted provided that wound healing has occurred, all complications have resolved, and at least 14 days have elapsed (for major operative procedures) prior to the tumor resection. * Patients of childbearing potential (or female partners of male participants) must be willing to take the appropriate precaution to avoid pregnancy or fathering a child for the duration of the study and practice an approved, highly effective method of birth control during treatment and for 12 months after their last dose of IL-2. Approved methods of birth control are as follows: * Combined (estrogen and progesterone containing) hormonal birth control associated with inhibition of ovulation: oral, intravaginal, transdermal * Progesterone-only hormonal birth control associated with inhibition of ovulation: oral, injectable, implantable * Intrauterine device (IUD) * Intrauterine hormone-releasing system (IUS) * Bilateral tubal occlusion * Vasectomized partner * True sexual abstinence when this is in line with the preferred and usual lifestyle of the patient. Periodic abstinence (eg, calendar ovulation, symptothermal, post-ovulation methods) is not acceptable * Patients (or legally authorized representative) must have the ability to understand the requirements of the study, have provided written informed consent as evidenced by signature on an ICF approved by an Institutional Review Board/Independent Ethics Committee (IRB/IEC), and agree to abide by the study restrictions and return to the site for the required assessments, including the OS Follow-up Period. Exclusion Criteria: * Patients who have received an organ allograft or prior cell transfer therapy that included a non-myeloablative or myeloablative chemotherapy regimen. * Patients who have a history of hypersensitivity to any component or excipient of LN-144/LN-145 or other study drugs: * NMA-LD preconditioning regimen (cyclophosphamide, mesna, and fludarabine) * Proleukin®, aldesleukin, IL-2 * Antibiotics (ABX) of the aminoglycoside group (i.e., streptomycin, gentamicin); except those who are skin-test negative for gentamicin hypersensitivity * Any component of the LN-144/LN-145 infusion product formulation including dimethyl sulfoxide (DMSO), human serum albumin (HSA), IL-2, and dextran-40. * Patients with symptomatic brain metastases (of any size and any number). o Patients with definitively treated brain metastases may be considered for enrollment, if, prior to tumor resection for TIL, the patient is clinically stable for ≥ 14 days, there are no symptomatic brain lesions, and that the patient does not require ongoing corticosteroid treatment. * Patients who are on chronic systemic immunosuppressive therapy except for those requiring steroid therapy for management of adrenal insufficiency; these patients may receive no more than 10 mg of prednisone or its equivalent daily. Transient use of steroids, e.g. in the perioperative period, is not an exclusion. * Patients who are pregnant or breastfeeding. * Patients who have active medical illness(es) that would pose increased risk for study participation, including: active systemic infections requiring systemic ABX, coagulation disorders, or other active major medical illnesses of the cardiovascular, respiratory, or immune systems. * Patients who have received a live or attenuated vaccination within 28 days prior to the start of NMA-LD pre-conditioning regimen. * Patients who have any form of primary immunodeficiency (such as severe combined immunodeficiency disease \[SCID\] and acquired immunodeficiency syndrome \[AIDS\]). * Patients who have a left ventricular ejection fraction (LVEF) \<45% or New York Heart Association (NYHA) functional classification \> Class 1. * Patients ≥ 60 years of age and who have a history of ischemic heart disease, chest pain, or clinically significant atrial and/or ventricular arrhythmias must have a cardiac stress test. * Patients with any irreversible wall movement abnormalities are excluded. * Patients who have a smoking history or signs or symptoms of obstructive or restrictive pulmonary disease and have a documented forced expiratory volume in 1 second (FEV1) of ≤ 60% of predicted normal: * If a patient is not able to perform reliable spirometry due to abnormal upper airway anatomy (i.e., tracheostomy), a 6-minute walk test may be used to assess pulmonary function. * Patients who are unable to walk a distance of at least 80% predicted for age and sex or demonstrates evidence of hypoxia at any point during the test (SpO2 \< 90%) are excluded. * Active, uncontrolled systemic infections, including COVID-19, within 30 days of surgery or NMA-LD. An uncomplicated bacterial UTI treated successfully with symptom resolution is not an exclusion. * Participation in another clinical study with an investigational product within 21 days of the initiation of NMA-LD. * No other active, concurrent malignancy that requires ongoing systemic treatment (e.g. indolent prostate) or interferes with radiographic assessment of response as determined by the investigator. Exceptions may allow for adjuvant NED cancers undergoing hormone-based therapy assuming the other eligibility criteria are met and the PI affirms the hormonal agent would not change the response. Eligibility Designation for Lymphodepletion Patients meeting eligibility criteria above between Day -21 and Day -8 prior to the planned initiation of lifileucel will be enrolled to the therapeutic portion of the protocol. All patients' eligibility criteria, including repeating cardiopulmonary function tests as necessary, will be reassessed within several days prior to the scheduled lymphodepletion in all cases. Prior to beginning the NMA-LD preparative regimen the following requirements must be met: * Patients must meet all eligibility criteria at the time of NMA-LD. * Full resolution of any active infection should be documented * Critical evaluation of changes in cardiovascular, respiratory, renal, coagulopathy, or immune problems and other major illnesses that may have developed or worsened must be documented * Patients with symptomatic, recurrent, pleural effusions that require drainage should not proceed to lymphodepletion without prior placement of a temporary in-dwelling pleural drain * All active medical issues must be addressed by Investigator or designee. Re-evaluation with additional imaging or testing may be required Benefit over risk should be assessed and reassessed throughout the treatment course. Successive lifileucel components (preparative lymphodepleting chemotherapy, LN-144/LN-145 and IL-2 ) should be withheld or discontinued if at any time during the treatment course, at the discretion of the treating physician, benefit may not be justified by risks to the patient Subsequent delays of lymphodepletion up to 14 days due to logistical issues such as production of lifileucel and/or major weather events will not constitute protocol violations and out of window assessments will not need to be repeated unless there is a change in clinical status.