Fully Human Bispecific Anti-CD20, Anti-CD19 CAR T Cells for Patients With Relapsed and/or Refractory B Cell Malignancies

General Inclusion Criteria for All Patients 1. Patients must be aged ≥ 18 years and ≤80 years with relapsed or refractory B-cell malignancy. 2. Absolute CD3 count ≥50 mm\^3. 3. Magnetic resonance imaging (MRI) brain and lumbar puncture with cerebral spinal fluid (CSF) analysis by cytology and flow cytometry without evidence of central nervous system (CNS) involvement ONLY in patients with: 1. A history of CNS involvement OR 2. A clinical suspicion at the time of enrollment. 4. Karnofsky performance score ≥70. 5. Adequate hepatic function, defined as aspartate aminotransferase (AST), alanine aminotransferase (ALT), and alkaline phosphatase \<3 × upper limit of normal (ULN); serum bilirubin \<2.0 mg/dL, or considered not clinically significant as per the clinical principal investigator's discretion (e.g., Gilbert's or indirect hyperbilirubinemia). 6. Absolute Neutrophil Count (ANC) ≥1000 with no Granulocyte Colony-Stimulating Factor (G-CSF) within 72 hours or pegylated G-CSF within 14 days. 7. Platelets ≥50,000 with no transfusion within 72 hours. 8. Adequate renal function, defined as creatinine clearance ≥50 mL/min measured by CKD-EPI 2021 calculator OR 24-hour urine collection. 9. Left ventricular ejection fraction of ≥45% -- by cardiac echocardiogram (ECHO) or Multigated Acquisition scan (MUGA) -- and adequate pulmonary function as indicated by room air oxygen saturation of ≥92%. 10. Expected survival \>12 weeks. 11. No contraindication to central line access. 12. Patient has demonstrated compliance with prior therapies. 13. Meet criteria for regarding fertility and contraception detailed below. 14. Negative urine or serum pregnancy test in females of childbearing potential at study entry. 15. Agree to practice birth control during the study. 16. Able to provide written informed consent. Disease-Specific Inclusion Criteria 1. Patients must meet at least one of the following criteria within the categories below: a. CAR/Bispecific antibody exposed: i. Relapsed after prior murine CD19 autologous CAR-T cell therapy and be \>90 days post prior CAR-T cell therapy OR relapsed after bispecific T-cell engaging therapy as a second or later line treatment. ii. \<5% presence of circulating CAR-T cells as measured by flow cytometry in patients with prior CD19 CAR T cell exposure. b. CAR naïve patients i. For Diffuse Large B-Cell Lymphoma (DLBCL): Progressed after two or more lines of therapy, including at a minimum CD20 antibody and combination cytotoxic chemotherapy regimen (e.g., CHOP, CHP, EPOCH, HyperCVAD) or relapse/progression after autologous stem cell transplant ii. For Chronic Lymphocytic Leukemia (CLL): Progressed after two or more lines of therapy, including both a covalent Bruton tyrosine kinase (BTK) inhibitor and a BCL2 inhibitor iii. For Mantle Cell Lymphoma (MCL): Progressed after two lines of therapy, including CD20 antibody, BTK inhibitor, and one cytotoxic chemotherapy regimen (e.g., bendamustine, cytarabine, CHOP) 2. Patients must have measurable disease defined as: 1. For B-cell NHL (DLBCL and MCL): Measurable disease must be documented within 4 weeks of the time of consent, defined as nodal lesions greater than 15 mm in the long axis or extranodal lesions \>10 mm in long and short axis OR bone marrow involvement that is biopsy-proven for B-cell Non-Hodgkin Lymphoma (NHL) 2. For CLL/ Small Lymphocytic Lymphoma (SLL): Indication for treatment as defined as any of the following: measurable lymph nodes ≥ 1.5 cm in the greatest transverse diameter and/or hepatomegaly or splenomegaly OR bone marrow involvement with ≥10% CLL involvement Exclusion Criteria for All Patients 1. Positive beta human chorionic gonadotropin (hCG) test in female of child-bearing potential. 2. Confirmed active human immunodeficiency virus (HIV), Hepatitis B or C infection. 3. History of significant autoimmune disease OR active, uncontrolled autoimmune phenomenon requiring steroid therapy defined as \>20 mg of prednisone or equivalent daily. 4. Presence of Grade ≥3 non-hematologic toxicities per National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE)version 5.0 unless believed to be due to underlying disease. 5. Concurrent use of investigational therapeutic agents or enrollment on another therapeutic clinical trial at any institution. A minimum of 14 days or 5 half-lives of the drug, whichever is shorter, of washout is required prior to apheresis. 6. Patients with active CNS involvement by malignancy on MRI or by lumbar puncture: a. Patients with prior CNS disease that have been effectively treated will be eligible, provided treatment was \>4 weeks before enrollment and a remission documented within 8 weeks of planned CAR-T cell infusion by MRI brain and CSF analysis. 7. Previous recipients of allogeneic hematopoietic stem cell transplantation (AHCT) are excluded if they are \<100 days post-transplant, have evidence of active graft-versus-host-disease (GVHD) of any grade, or are currently on immunosuppression. 8. Anti-CD20 antibody treatment within 4 weeks of cell infusion. 9. Anti-CD19 antibody treatment within 4 weeks of cell infusion. 10. Cytotoxic chemotherapy other than lymphodepletion within 14 days of the lymphodepletion start date. 11. Cytotoxic chemotherapy treatment within 14 days or steroid treatment (other than replacement dose steroids) within 7 days prior to apheresis collection for CAR T cells. 12. Oral chemotherapeutic agents or antibody-directed treatment within 7 days of apheresis: a. BTK inhibitors are allowed until 1 day prior to apheresis and can be restarted until 1 day prior to lymphodepletion. 13. Patients post solid organ transplant who develop high-grade lymphomas or leukemias. 14. Concurrent active malignancy other than basal or squamous cell carcinomas of the skin (underlying low-grade lymphoma chronic lymphocytic leukemia/FL/MZL is allowable in patients with transformed large cell lymphoma). 15. Refusal to participate in the long-term follow-up protocol. Special Criteria Regarding Fertility and Contraception Female subjects of reproductive potential (women who have reached menarche or women who have not been post-menopausal for at least 24 consecutive months, i.e., who have had menses within the preceding 24 months, or have not undergone a sterilization procedure \[hysterectomy or bilateral oophorectomy\]) must have a negative serum or urine pregnancy test performed as part of the eligibility criteria. Due to the high-risk level of this study, while enrolled, all subjects must agree not to participate in a conception process (e.g., active attempt to become pregnant or to impregnate, sperm donation, in vitro fertilization). Additionally, if participating in sexual activity that could lead to pregnancy, the study subject must agree to use a reliable and double-barrier methods of contraception during the follow-up period of the protocol. Acceptable birth control includes a combination of two of the following methods: * Condoms (male or female) with or without a spermicidal agent. * Diaphragm or cervical cap with spermicide. * Intrauterine device (IUD). * Hormonal-based contraception. Subjects who are not of reproductive potential (women who are premenarche or have been post-menopausal for at least 24 consecutive months or have undergone hysterectomy, tubal ligation, salpingectomy, and/or bilateral oophorectomy, or men who have documented azoospermia) are eligible without requiring the use of contraception.