Umbilical Cord Blood Immune Cells and Chemotherapy in Treating Participants with Recurrent or Refractory CD19 Positive Acute Lymphoblastic Leukemia, Chronic Lymphocytic Leukemia, or Non-Hodgkin Lymphoma

Inclusion Criteria

• Patients with history of CD19 positive B-lymphoid malignancies, defined as acute lymphoblastic leukemia (ALL), diffuse large B-cell Lymphoma (DLBCL), chronic lymphocytic leukemia (CLL) small lymphocytic lymphoma (SLL), primary mediastinal large B-cell lymphoma, Richter’s transformation of CLL or SLL, follicular lymphoma, marginal zone lymphoma and high grade transformation of follicular or marginal zone lymphoma who have received at least 2 lines of standard chemoimmunotherapy or targeted therapy and have persistent disease
• Patients with CD19 positive B-lymphoid malignancies as defined above with relapsed disease following standard therapy or a stem cell transplant.
• Patients at least 3 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.
• Karnofsky/Lansky performance scale > 70.
• Serum creatinine =< 1.5 mg/dL or estimated glomerular filtration rate (eGFR using the Chronic Kidney Disease Epidemiology Collaboration [CKI-EPI] equation) >= 60 ml/min/1.73 m^2
• Alanine aminotransferase (ALT)/aspartate aminotransferase (AST) =< 2.5 x upper limit of normal (ULN) or =< 5 x ULN if documented liver metastases.
• Total bilirubin =< 1.5 mg/dL, except in subjects with Gilbert’s Syndrome in whom total bilirubin must be =< 3.0 mg/dL
• No history of liver cirrhosis.
• No ascites.
• Cardiac ejection fraction >= 50%, no clinically significant pericardial effusion as determined by an echocardiogram (ECHO) or multiple-gated acquisition (MUGA), and no clinically significant electrocardiography (ECG) findings. Clinical significance attributed per principal investigator (PI) discretion
• No clinically significant, per PI discretion, pleural effusion, baseline oxygen saturation > 92% on room air
• Able to provide written informed consent.
• 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.
• Signed consent to long-term follow-up protocol PA17-0483.

Exclusion Criteria

• Positive beta human chorionic gonadotropin (HCG) in female of child-bearing potential defined as not postmenopausal for 24 months or no previous surgical sterilization or lactating females.
• Known positive serology for human immunodeficiency virus (HIV).
• Presence of clinically significant grade 3 or greater toxicity from the previous treatment, as determined by PI.
• Presence of fungal, bacterial, viral, or other infection requiring IV antimicrobials for management. Note: Simple urinary tract infection (UTI) and uncomplicated bacterial pharyngitis are permitted if responding to active treatment.
• Presence of active neurological disorder(s).
• Concomitant use of other investigational agents.
• Previously received any anti-CD19 therapy (for example, chimeric antigen receptor [CAR] treatment or blinatumomab).