Genetically Engineered Cells (BAFFR-CAR T Cells) for the Treatment of Relapsed or Refractory B-cell Acute Lymphoblastic Leukemia and B-cell Lymphoblastic Lymphoma

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative * Agreement to allow the use of archival tissue from diagnostic tumor biopsies * If unavailable, exceptions may be granted with study principal investigator (PI) approval * Age ≥ 18 years * Eastern Cooperative Oncology Group (ECOG) ≤ 2 * Life expectancy ≥ 16 weeks * Histologically confirmed B-ALL or B-cell lymphoblastic lymphoma * Relapsed/refractory disease. Minimal residual disease (MRD) relapse is allowed * Evidence of tumor expressing BAFF-R at any level either by flow or immunohistochemistry * Recovered to ≤ grade 1 from the acute toxic effects (except alopecia and peripheral neuropathy) of prior anti-cancer therapy * No known contraindications to leukapheresis, steroids or tocilizumab * Ineligible for or failed prior CD19-targeted immunotherapy (e.g., blinatumomab or CD19-CAR T cells) * For participants who had prior CD19-CAR T cell therapy: * At least 90-days has elapsed since participant received last CD19-CAR T cell therapy AND * Persistence of prior CD19-CAR T cells must be evaluated and found to be \< 5% prior to leukapheresis procedure * Note: Participants who have undergone stem cell transplantation (at least 100 days prior to enrollment) after the last CD19-CAR T cell therapy, are considered eligible and do not need to meet the above criteria * Participants with central nervous system (CNS) involvement by leukemia (CNS2 and asymptomatic CNS3) may be considered eligible after discussions with the study team * Total serum bilirubin ≤ upper limit of normal (ULN) (unless has Gilbert's disease or related to liver involvement by leukemia, then ≤ 3.0) * Aspartate aminotransferase (AST) ≤ ULN, unless related to liver involvement by ALL, then ≤ 3.0 * Alanine aminotransferase (ALT) ≤ ULN, unless related to liver involvement by ALL, then ≤ 3.0 * Creatinine clearance of ≥ 40 mL/min per 24-hour urine test or the Cockcroft-Gault formula * Left ventricular ejection fraction (LVEF) ≥ 50% * Oxygen (O2) saturation ≥ 92% on room air * Seronegative for HIV quantitative polymerase chain reaction (qPCR), hepatitis C virus (HCV), and active hepatitis B virus (HBV) (surface antigen negative), and syphilis (rapid plasma reagin \[RPR\]) * If positive, hepatitis C ribonucleic acid (RNA) quantitation must be performed OR * If seropositive for HIV, HCV or HBV, nucleic acid quantitation must be performed. The viral load must be undetectable * Meets other institutional and federal requirements for infectious disease titer requirements * Note Infectious disease testing to be performed within 28 days prior to start of protocol therapy * Women of childbearing potential (WOCBP): Negative urine or serum pregnancy test * If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * QuantiFERON-tuberculosis (TB) Gold or equivalent * Results do not impact patient eligibility; however, the test must be initiated prior to enrollment * Agreement by females and males of childbearing potential to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 3 months after the last dose of protocol therapy * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) * A complete liver evaluation (which includes ultrasound elastography, MRI of the liver and a hepatology consult) may be done if needed based on PI's recommendation Exclusion Criteria: * Autologous/allogeneic stem cell transplant within 100 days at the time of enrollment * Immunosuppressant medications within 1 month prior to protocol enrollment * Concurrent use of systemic steroids or chronic use of immunosuppressant medications. Recent or current use of inhaled steroids is not exclusionary. Physiologic replacement of steroids (prednisone ≤ 7.5 mg /day or equivalent) is allowed * Auto-immune disease or active graft-versus-host disease (GvHD) within 3 months prior to protocol enrollment requiring systemic immunosuppressant therapy * Class III/IV cardiovascular disability according to the New York Heart Association (NYHA) Classification * Subjects with clinically significant arrhythmia or arrhythmias not stable on medical management within 2 weeks of enrollment * Any abnormal liver enzyme levels (as defined by grade 1 elevation from ULN in ALT, AST, and bilirubin levels) at time of enrollment, unless they are abnormal due to liver involvement by leukemia per the treating physician's discretion * Subjects with a known history or prior diagnosis of uncontrolled central nervous system (CNS) disorders such as optic neuritis or other immunologic or inflammatory disease affecting the CNS, including uncontrolled seizure disorder * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agent * Known significant bleeding disorders (e.g., severe von Willebrand's disease) or hemophilia * History of venous occlusive disease (VOD), or GvHD * Subjects with a history of the following GvHD may still be included in the study: * Resolved grade 2 or less steroid-sensitive acute skin GvHD * Grade 1 gastrointestinal (GI)-GvHD developed within 100 days post prior alloHCT * Limited chronic GVHD * History of stroke or intracranial hemorrhage within 6 months of enrollment * History of other malignancies, except for malignancy surgically resected (or treated with other modalities) with curative intent, basal cell carcinoma of the skin or localized squamous cell carcinoma of the skin; non-muscle invasive bladder cancer; malignancy treated with curative intent with no known active disease present for ≥ 2 years * Clinically significant uncontrolled illness * Active systemic uncontrolled infection * Known history of immunodeficiency virus (HIV) or hepatitis B or hepatitis C infection * Females only: Pregnant or breastfeeding * Any other condition that would, in the investigator's judgment, contraindicate the subject's participation in the clinical study due to safety concerns with clinical study procedures * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)