Inclusion Criteria: * Age \>= 18 years * Relapsed or refractory CD19+ B cell malignancies of the one of the following histopathology: * Biopsy proven B-cell non-Hodgkin lymphoma (NHL) of any histopathology (including Richter Transformation of CLL); relapsed or refractory disease defined as: * Two or more prior lines of therapy, at least one anthracycline containing regimen, unless intolerable. Exception: Patients with Richter transformation of CLL are eligible if they had \>= one prior treatment, including prior BTK inhibition * Demonstration of progressive or stable disease by positron emission tomography/computed tomography (PET/CT) or CT criteria as the best response to the most recent chemotherapy regimen according to the revised Lugano Response Criteria for Malignant Lymphoma. * Measurable disease defined as measurable by CT portion of a PET/CT: To be considered measurable, the must be at least one lesion that has a single diameter of (\>1.5 cm Note: Lesions that have been previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy * Biopsy proven SLL or flow cytometry proven CLL; relapsed disease defined as: * \>= two prior lines of therapy, and/or \>= 6 months of second line prior BTK inhibition (e.g. venetoclax and ibrutinib). Exception: Patients in stable disease (SD) or partial response (PR) with a known ibrutinib resistance mutation (BTK or phospholipase Cgamma2) may be included even if on ibrutinib therapy for less than 6 months. * Demonstration of progressive or stable disease by PET/CT or CT criteria according to the International Workshop on Chronic Lymphocytic Leukemia (iwCLL2018) criteria * Measurable disease by CT portion of a PET/CT where at least one lesion has a single diameter of \>1.5 cm or peripheral blood absolute blood lymphocyte count (ALC) of \> 5000. Note: Lesions that have been previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * Hemoglobin \>= 8.0 g/dL (=\< 14 days prior to registration) * Absolute neutrophil count (ANC) \>= 500/mm\^3 (=\< 14 days prior to registration) * Platelet count \>= 30,000/mm\^3 (=\< 14 days prior to registration) * Total bilirubin =\< 2.0 mg/dL (with the exception of subjects with Gilbert's syndrome. Subjects with Gilbert's syndrome may be included if their total bilirubin is =\< 3.0 x upper limit of normal (ULN) and direct bilirubin =\< 1.5 x ULN) (=\< 14 days prior to registration) * Alanine aminotransferase (ALT) and aspartate transaminase (AST) =\< 3 x ULN (=\< 14 days prior to registration) * Prothrombin time (PT) / international normalized ratio (INR) and/or activated partial thromboplastin time (aPTT) =\< 1.5 x ULN OR if patient is receiving anticoagulant therapy and INR or aPTT is within target range of therapy (for patients receiving anticoagulation, there should be no prior history of bleeding, and no recent deep venous thrombosis/pulmonary embolism (DVT/PE) within the last 6 months of enrollment) (=\< 14 days prior to registration) * Calculated creatinine clearance \>= 45 ml/min using the Cockcroft-Gault formula (=\< 14 days prior to registration) * Cardiac ejection fraction \>= 50% and no evidence of clinically significant pericardial effusion as determined by an echocardiogram (ECHO) or multigated acquisition scan (MUGA) scan * Baseline oxygen saturation \>= 92% on room air * Negative serum pregnancy test done =\< 7 days prior to registration, for persons of childbearing potential only * Women patients of child bearing potential, including women with tubal ligations, must commit to using use 2 highly effective forms of birth control (defined as the use of an intrauterine device, a barrier method with spermicide, condoms, any form of hormonal contraceptives) for the duration of the study and for 12 months following IC19/1563 therapy * Provide written informed consent * Willingness to provide mandatory blood specimens for correlative research * Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study) Exclusion Criteria: * Any of the following because this study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown: * Pregnant persons * Nursing persons * Women of childbearing potential who are unwilling to employ highly effective contraception * Sexually active males who are not willing to use contraception during the study and for \>= 12 months after IC19/1563 therapy * Patients who are able to obtain market approved CD19 CAR T-cell therapies * Live vaccine =\< 6 weeks prior to start of registration * Autologous stem cell transplant =\< 6 weeks of registration * History of allogenic stem cell transplant if was performed less than 100 days prior to registration, if patients have active graft-versus host disease (GVHD) or are if patients are on chronic immunosuppression. Patients with allogeneic transplantation more than 100 days prior to registration, with no active GVHD and who are not on immunosuppression are eligible * History of a seizure disorder, cerebrovascular ischemia/hemorrhage, dementia, cerebellar disease, or any autoimmune disease with central nervous system (CNS) involvement * Any form of primary immunodeficiency such as severe combined immunodeficiency disease * Current need of systemic corticosteroid therapy, in doses over 20 mg /day of prednisone or equivalent forms of steroids * History of severe immediate hypersensitivity reaction to CART19, stem cell infusion dimethyl sulfoxide (DMSO) or any of the CAR-T cryopreservation ingredients * History of malignancy other than non-melanoma skin cancer, carcinoma in situ (e.g. cervix, bladder, breast) or early stage cancers (Stage I or II), unless disease free for \>= 2 years * Clinically significant active infection (e.g. simple urinary tract infection \[UTI\], bacterial pharyngitis allowed) or currently receiving IV antibiotics or have received IV antibiotics =\< 7 days prior to registration. Note: prophylactic antibiotics, antivirals and antifungals are permitted * Known history of human immunodeficiency virus (HIV) infection or acute or chronic hepatitis B or hepatitis C infection. Subjects with a history of hepatitis infection must have cleared their infection as determined by standard serological and genetic testing per current Infectious Diseases Society of America (IDSA) guidelines. Prophylactic antiviral therapy should be considered per institutional guidelines * History of any of the following cardiovascular conditions =\< 6 months: * Class III or IV heart failure as defined by the New York Heart Association (NYHA) * Cardiac angioplasty or stenting * Myocardial infarction * Unstable angina * Or other clinically significant cardiac disease * Any other acute or chronic medical or psychiatric condition that may increase the risk associated with study participation or investigational product administration or that, in the judgment of the investigator, would make the subject inappropriate for entry into the study * Concurrent cancer therapy. The following are exceptions: * Treatment with therapies may continue at time of registration; however, the washout period must be met prior to leukapheresis * Treatment with any other investigational agent may continue at time of registration provided last date of treatment is =\< 14 days prior to leukapheresis.