UB-VV400 in Combination With Rapamycin in Relapsed or Refractory B-cell Malignancies

Inclusion Criteria: 1. Age ≥ 18 at time of consent. 2. Provide voluntary written informed consent. 3. Relapsed/refractory disease for subjects that are either CAR T-naive or CAR T-exposed. 4. Measurable disease according to Lugano 2014 criteria. 5. No serious concomitant diseases or active/uncontrolled infections. 6. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. 7. Cardiac function: left ventricular ejection fraction (LVEF) ≥ 40%. 8. Pulmonary function: pulse oximetry ≥ 90% on room air at rest. 9. Renal function: serum creatinine ≤ 1.5 × age-adjusted upper limit of normal (ULN) or creatinine clearance ≥ 45 mL/min. 10. Absolute lymphocyte count (ALC) ≥ 0.2×10\^9/L. 11. Alanine aminotransferase (ALT) ≤ 2.5 × ULN, aspartate aminotransferase (AST) ≤ 2.5 × ULN, AND total bilirubin \< 1.5 × ULN. 12. No ongoing coagulopathies requiring periodic replacement of clotting factors (eg, fresh frozen plasma, cryoprecipitate). 13. Women of childbearing potential must: 1. Have 2 negative pregnancy tests verified (one negative serum beta human chorionic gonadotropin \[β-hCG\] at screening and another within 48 hours prior to treatment withUB-VV400). 2. Commit to "true abstinence" from heterosexual intercourse or agree to use and complywith highly effective, uninterrupted contraception for 12 months after administration of UB-VV400 . 3. Abstain from breastfeeding for 12 months following administration of UB-VV400. 14. Men with partners of childbearing potential must commit to "true abstinence" from heterosexual intercourse or agree to use a highly effective form of contraception duringheterosexual contact with a pregnant individual or any individual of childbearing potential for 12 months after administration of UB-VV400, regardless of past vasectomy. 15. Subjects must agree not to donate blood, organs, sperm/semen, and/or egg cells for use for at least 1 year following treatment with UB-VV400 alone or in combination with rapamycin. Insufficient data are available to define a duration of time sufficient to make a recommendation on when it is safe to donate any tissue; therefore, subjects should not donate any tissue after administration of UB-VV400. Exclusion Criteria: 1. Women who are pregnant or breastfeeding. 2. Subjects with current isolated central nervous system (CNS) tumor involvement. 3. Subjects with a prior malignancy whose clinical course or management has the potential to interfere with the safety and/or efficacy evaluation of the clinical trial. 4. Prior treatment with any of the following: allogeneic bone marrow transplantation, gene therapy, or adoptive cell transfer of any kind except for CAR T-cell therapy. 5. Treatment with prior CD22-directed therapy except for UB-VV400. 6. History of or active human immunodeficiency virus (HIV). 7. Active hepatitis B (HepB) or hepatitis C (HepC). 8. For subjects receiving rapamycin: a. History of angioedema; b. Pneumonitis (Grade 3 or greater). 9. Ongoing Grade \> 2 toxicities from the last line of anticancer therapy. 10. Use of the following: 1. Therapeutic systemic doses of corticosteroids (defined as \> 20 mg prednisone equivalent) within 72 hours before dosing with UB-VV400. 2. Approved targeted therapies: i. Small molecules: within 3 half-lives before dosing with UB-VV400 (see package insert). ii. Antibodies: within 14 days before dosing with UB-VV400. iii. CAR T therapy: within 28 days before dosing with UB-VV400. c. Autologous stem cell transplant within 28 days before dosing with UB-VV400. d. Cytotoxic chemotherapy (eg, alkylators, anthracyclines) within 14 days before dosing with UB-VV400. e. Any experimental agent (ie, not approved for disease/indication or with accepted consensus guidelines recommendation) within 4 weeks before dosing with UB-VV400 unless progression has been documented and a minimum of 3 half-lives has elapsed. f. Any immune-suppressing agent within 28 days before dosing with UB-VV400 (eg, tacrolimus, mycophenolate mofetil, immunosuppressive antibodies such as anti-tumor necrosis factor \[TNF\]/IL-6). g. Radiation within 4 weeks before dosing with UB-VV400 (palliative radiation to a symptomatic lesion\[s\] is allowed if at least one additional, non-irradiated, measurable lesion remains present to assess response). h. Prophylactic treatment with short-acting oral antiretroviral medications within 7 days before UB-VV400 administration. Long-acting antiretroviral prophylaxis is not allowed within 2 years of UB-VV400 treatment. 11. Allergies to rapamycin or supportive medications required for CAR T-cell toxicity management (eg, tocilizumab). 12. Systemic autoimmune diseases or immunodeficiency diseases (except for well-controlled type I diabetes with hemoglobin A1C \[HbA1c\] less than 8% or well-controlled thyroid disease, as assessed by the treating physician). 13. Ongoing CNS diseases (eg, seizure disorder, tremor, history of cerebral vascular accident \[CVA\]/recurrent transient ischemic attack \[TIA\]) that would preclude evaluation of immune effector cell-associated neurotoxicity syndrome (ICANS). 14. Presence of uncontrolled angina or other acute uncontrolled heart disease. Myocardial infarction within the previous 6 months. New York Heart Association (NYHA) Class III or IV. History of any of the following conditions: syncope of cardiovascular etiology, ventricular arrhythmia of pathological origin (including, but not limited to, ventricular tachycardia and ventricular fibrillation), or sudden cardiac arrest. 15. Actively receiving treatment in other interventional clinical trials. Note: continued follow-up on previous trials is allowed for survivorship, but no further investigational agents or assessments will be allowed.