A Study to Find a Suitable Dose of ASP2802 in People With CD20-positive B-cell Lymphomas

Inclusion Criteria: * Participant has histologically-confirmed cluster of differentiation (CD) 20-positive B-cell lymphoma (CD20 expression should be demonstrated by local testing at the time of relapse or progression and within 1 month of study screening) which may include the following aggressive large cell or indolent subtypes. * Diffuse large B-cell lymphoma (DLBCL), Not otherwise specified (NOS) * Follicular lymphoma grade 3B * Primary mediastinal large B-cell lymphoma * T-cell/histiocyte-rich large B-cell lymphoma * DLBCL associated with chronic inflammation * Intravascular large B-cell lymphoma * Anaplastic lymphoma kinase fusion gene (ALK)+ large B-cell lymphoma * B-cell lymphoma, unclassifiable, with features intermediate between DLBCL and classical Hodgkin lymphoma * High grade B-cell lymphoma with proto-oncogene located on chromosome 8 (MYC) and B-cell leukemia/lymphoma 2 gene (BCL2) and/or B-cell leukemia/lymphoma 6 gene (BCL6) rearrangements * High-grade B-cell lymphoma, NOS * Human herpes virus type 8 (HHV8)+ DLBCL, NOS * Primary cutaneous DLBCL, leg type * Transformed follicular lymphoma * Follicular lymphoma * Marginal zone lymphoma * Participant has either relapsed or refractory disease defined as: * Relapse after ≥ 2 prior lines of therapy; or * Refractory disease: After ≥ 2 prior lines of therapy and did not achieve complete response (CR) * Participants who have undergone autologous stem cell transplant (SCT) with disease progression or relapse following SCT will be eligible if all other eligibility criteria are met. * Participants with lymphoma must have evaluable or measurable disease according to the Lugano criteria for response assessment of lymphoma. Lesions that have been previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy. * At least 2 weeks or 5 half-lives, whichever is longer, must have elapsed since any prior anticancer therapy at the time of leukapheresis, except for inhibitory/stimulatory immune checkpoint therapy, which requires 3 half-lives. Exceptions: * There is no time restriction with regard to prior intrathecal chemotherapy (including steroids) provided there is complete recovery from any acute toxic effects of such. * Participants receiving steroid therapy at physiologic replacement doses (≤ 5 milligrams per day (mg/day) of prednisone or equivalent doses of other corticosteroids) are allowed. Use of higher doses of systemic steroids must stop at least 72 hours before leukapheresis. * Radiation therapy must have been completed at least 2 weeks prior to leukapheresis. * Toxicities due to prior therapy must be stable and recovered to ≤ Grade 1 (except for clinically non-significant toxicities, such as alopecia, nutritional support measures, electrolyte abnormalities or those not impacting the investigator's ability to assess treatment-emergent toxicities). * Participant has cardiac ejection fraction (EF) ≥ 45%, no evidence of physiologically significant pericardial effusion as determined by an echocardiogram or multigated acquisition scan (MUGA) and no clinically significant ECG findings. * Participant has no clinically significant pleural effusion. * Participant has baseline oxygen saturation \> 90% on room air at rest. * Participant has a life expectancy of ≥ 12 weeks. * Participant has an ECOG performance score of 0, 1 or 2. * Female participant is not pregnant and at least 1 of the following conditions apply: * Not a woman of childbearing potential (WOCBP). * WOCBP who agrees to follow the contraceptive guidance from the time of informed consent through at least 6 months after final study treatment administration. * Female participant must agree not to breastfeed starting at screening and throughout the study period and for 6 months after final study treatment administration. * Female participant must not donate ova starting at first dose of IP and throughout the study period and for 6 months after final study treatment administration. * Male participant with female partner(s) of childbearing potential (including breastfeeding partner) must agree to use contraception throughout the treatment period and for 6 months after final study treatment administration. * Male participant must not donate sperm during the treatment period and for 6 months after final study treatment administration. * Male participant with pregnant partner(s) must agree to remain abstinent or use a condom for the duration of the pregnancy throughout the study period and for 6 months after final study treatment administration. * Participant agrees not to participate in another interventional study while participating in the present study * Participant has adequate organ function at baseline. If a participant has received a recent blood transfusion, the laboratory tests must be obtained ≥ 28 days after any blood transfusion. Additional Inclusion Criteria prior to Lymphodepletion Chemotherapy: * Participant has adequate hepatic function defined by aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3 × upper limit of normal (ULN); total bilirubin ≤ 1.5 × ULN (participants with Gilbert's syndrome may be included if their total bilirubin is ≤ 3.0 × ULN and direct bilirubin is ≤ 1.5 × ULN) * Participant has adequate renal function defined by: * Creatinine \< 1.5 × ULN, or * An estimated creatinine clearance (CrCl) of ≥ 60 milliliter per minute (ml/min) as calculated by the Cockcroft-Gault equation * Participant has adequate bone marrow function defined by absolute neutrophil count ≥ 500/microliter (µl) and platelet count ≥ 50 000/µl (unless inadequate bone marrow function is thought to be related to bone marrow involvement with the lymphoproliferative neoplasm). Exclusion Criteria: * Participant has been diagnosed with the following conditions: * B-lymphoblastic leukemia/lymphoma (B-ALL/LBL) * chronic lymphocytic leukemia (CLL)/small lymphocytic lymphoma (SLL) * Burkitt lymphoma * Richter's transformation of CLL/SLL * Epstein-Barr virus (EBV)+ DLBCL, NOS * Mantle cell lymphoma * Known history or suspicion of central nervous system involvement by lymphoma. * Participant weighs \< 45 kilograms (kg) at screening. * Participant had prior allogeneic hematopoietic stem cell transplantation (HSCT). * Participant has received any investigational therapy within 28 days or 5 half-lives, whichever is longer, prior to screening. * Participant has a history of non-study related malignancy other than non-melanoma skin cancer or carcinoma in situ (e.g., cervix, bladder, breast) unless disease free for at least 3 years. * Participant has a history of myocardial infarction, cardiac angioplasty or stenting, unstable angina or other clinically significant cardiac disease within 12 months of enrollment or has cardiac atrial or cardiac ventricular lymphoma involvement. * Participant has known abnormal lung function: forced expiratory volume (FEV) \< 60% prediction, diffusing capacity of the lung for carbon monoxide \< 60% prediction, blood oxygen saturation \< 90% on room air. * Participant has intracranial hypertension or unconsciousness, respiratory failure or disseminated intravascular coagulation. * Participant has leukocytosis (white blood cell \[WBC\] count ≥ 25 000/µL) or rapidly progressive disease that would compromise ability to complete study therapy. * Participant has a history or presence of a central nervous system (CNS) disorder such as cerebrovascular ischemia/hemorrhage, dementia, cerebellar disease, seizure disorder or autoimmune disease with CNS involvement that may impair the ability to evaluate neurotoxicity. * Participant has any of the following per screening serology test: * Hepatitis A virus (HAV) antibodies immunoglobulin M (IgM) * Positive hepatitis B surface antigen (HBsAg) or detectable hepatitis B Deoxyribonucleic acid (DNA). Participants with negative HBsAg, positive hepatitis B core antibody (anti-HBc) and negative anti-HBs are eligible if hepatitis B DNA is undetectable * hepatitis C virus (HCV) antibodies unless HCV ribonucleic acid (RNA) is undetectable * Participant has human immunodeficiency virus (HIV) per screening serology test. * Participant has a positive severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) polymerase chain reaction (PCR) test at Screening. * Participant has a primary immunodeficiency or history of systemic autoimmune disease (e.g., Crohn's, rheumatoid arthritis, systemic lupus) resulting in end organ injury or requiring systemic immunosuppression/disease-modifying agents within the last 2 years. * Participant has presence of fungal, bacterial (e.g., tuberculosis), viral or other infection that is uncontrolled or requiring IV antimicrobials for management. Simple infections like urinary tract infections and uncomplicated bacterial pharyngitis are permitted if responding to active treatment. * Participant's adverse reactions (excluding alopecia or vitiligo) from prior therapy have not improved to Grade 1 or baseline at least 14 days prior to Screening. * Participant has received rituximab within 14 days prior to Screening. * Participant has presence of any other concurrent medical condition or factors that would prevent the participant from undergoing or completing protocol-specified treatment, or makes the participant unsuitable for study participation. * Participant had prior treatment with CD20 CAR-T cell or CD20 bi-specific therapy. * Participant has a known or suspected hypersensitivity to ASP2802 (MACT), MA-20, fludarabine, cyclophosphamide or any components of the formulations used. * Participant is receiving treatment with anticoagulants. Additional Exclusion Criteria required for Lymphodepletion Chemotherapy: * Participant received bridging therapy within 10 days prior to start of lymphodepletion chemotherapy (LDC). * Participant has an active systemic infection present or onset of fever ≥ 38°C/100.4°F, not related to underlying disease. Participants who meet either of these on the day of scheduled ASP2802 (MACT) infusion should have ASP2802 (MACT) administration delayed. * Participant has a positive SARS-CoV-2 PCR test. * Participant has an intercurrent illness or toxicity that would place the participant at undue risk of proceeding to LDC and ASP2802 (MACT) infusion. * Participant is receiving therapeutic doses of corticosteroids (defined as ≥ 20 mg/day prednisone or equivalent) within 24 hours prior to LDC. Physiologic replacement, topical, intranasal and inhaled steroids are permitted. * Participant has signs of pre-existing CNS disease or toxicity. * Participant has had major surgery within 28 days prior to the start of study treatment. * Participants should not experience a significant worsening in clinical status compared to initial eligibility criteria that would increase the risk of adverse events associated with LDC or ASP2802 (MACT) infusion.