Phase I/II Study of CD5 CAR Engineered IL15-Transduced Cord Blood-Derived NK Cells in Conjunction With Lymphodepleting Chemotherapy for the Management of Relapsed/Refractory Hematological Malignances

Inclusion criteria 1. Patients with hematological malignances with an expression of CD5 in the pre-enrollment tumor sample ≥ 30% measured by immunohistochemistry or flow cytometry. 2. Patients must meet diseases specific eligibility criteria (see below) 3. Patients should be at least 1 weeks from last cytotoxic chemotherapy at the time of starting lymphodepleting chemotherapy. Patients may continue tyrosine kinase inhibitors or other targeted therapies until at least three days prior to administration of lymphodepleting chemotherapy. 4. Localized radiotherapy to one or more disease sites are allowed prior the infusion provided that there are additional disease sites that are not irradiated. 5. Karnofsky Performance Scale \> 50%. 6. Adequate organ function: 1. Renal: Serum creatinine ≤ 2.0ULN or estimated Glomerular Filtration Rate (eGFR using the CKI-EPI equation) ≥ 30 ml/min/1.73 m2. 2. Hepatic: ALT/AST ≤ 3.0 x ULN or ≤ 5 x ULN if documented liver involvement with disease, Total bilirubin ≤ 2.0ULN, except in subjects with Gilbert's Syndrome in whom total bilirubin must be ≤ 3.0 mg/dL. No history of liver cirrhosis. 3. Cardiac: Cardiac ejection fraction ≥ 40%, no clinically significant pericardial effusion as determined by an ECHO or MUGA, and no uncontrolled arrhythmias or symptomatic cardiac disease. 4. Pulmonary: No clinically significant lung involvement, per PI discretion, pleural effusion, baseline oxygen saturation \> 92% on room air. 7. Able to provide written informed consent. 8. 18-80 years of age. 9. Weight ≥40 kg. 10. English and non-English speaking patients are eligible. 11. All participants who are able to have children must practice effective birth control while on study and up to 3 months post completion of study therapy. Acceptable forms of birth control for female patients include hormonal birth control, intrauterine device, diaphragm with spermicide, condom with spermicide, or abstinence, for the length of the study. If the participant is a female and becomes pregnant or suspects pregnancy, she must immediately notify her doctor. If the participant becomes pregnant during this study, she will be taken off this study. Men who are able to have children must use effective birth control while on the study. If the male participant fathers a child or suspects that he has fathered a child while on the study, he must immediately notify his doctor. 12. Signed consent to long-term follow-up protocol PA17-0483. 13. Disease specific inclusion criteria A. T-cell non-Hodgkin's lymphoma and T-cell acute lymphoblastic leukemia 1. Patients with history of T-lymphoid malignancies, defined as acute lymphoblastic leukemia (ALL/T-LBL), Peripheral T-cell lymphoma (PTCL-NOS), MF/SS, Hepatosplenic gamma/delta NHL, AITL, ALCL, or other subtypes of T cell NHL, T-PLL, Mixed phenotypic leukemia (MPAL) with CD5 expression who have received at least 2 lines of standard chemo-immunotherapy or targeted therapy and have measurable persistent disease. For T-ALL active disease defined as (\>5% of blasts or positive MRD at a level of \>0.1% measured by multiparameter flow cytometry). 2. Patients with history of T-lymphoid malignancies as defined above with relapsed disease following standard therapy or a stem cell transplant. B. Chronic lymphocytic leukemia (CLL) Chronic lymphocytic leukemia (CLL) small lymphocytic lymphoma (SLL), Richter's transformation of CLL or SLL who have received at least 2 lines of standard therapy or targeted therapy to include chemoimmunotherapy e.g. FCR, BTK inhibitors and a BCL-2 inhibitor and have persistent disease. C. Mantle cell lymphoma Relapsed or refractory mantle cell lymphoma after 2 lines of standard chemoimmunotherapy including a BTKi. Exclusion criteria: 1. Positive beta HCG in female of child-bearing potential defined as not postmenopausal for 24 months or no previous surgical sterilization or lactating females. 2. Presence of clinically significant Grade 3 or greater toxicity from the previous treatment, as determined by PI. 3. Presence of uncontrolled fungal, bacterial, viral, or other infection not responding to appropriate therapy. 4. Active hepatitis B or C. 5. HIV with detectable viral load. 6. Presence of active neurological disorder(s). 7. Active autoimmune disease within 12 months of enrollment 8. Active cerebral or meningeal involvement by the malignancy 9. Active (defined as requiring therapy) acute or chronic GVHD. 10. Any other malignancy known to be active, except for treated cervical intra-epithelial neoplasia and non-melanoma skin cancer. 11. Presence of any other serious medical condition that may endanger the patient at investigator criteria. 12. Major surgery \<4 weeks prior to first dose of study drug 13. Allogeneic SCT or DLI \<12 weeks prior to first dose of study drug. Recipients of an allogeneic SCT patients should have discontinued all forms of immunosuppression at least 8 weeks prior enrollment in the study. 14. Concomitant use of other investigational agents. 15. Concomitant use of other anti-cancer agents. 16. Patients receiving systemic steroid therapy at time of NK cell infusion (physiological substitutive doses are allowed) or have received ATG or lymphocyte immune globulin within 14 days or alemtuzumab within 3 months of enrollment. 17. Patients receiving immunosuppressive therapy. 18. Patients with diminished mental capacity will not be enrolled on the study.