This phase I trial studies the side effects and best dose of genetically modified T-cell therapy in treating patients with receptor tyrosine kinase-like orphan receptor 1 positive (ROR1+) chronic lymphocytic leukemia (CLL), mantle cell lymphoma (MCL), acute lymphoblastic leukemia (ALL), stage IV non-small cell lung cancer (NSCLC), or triple negative breast cancer (TNBC) that has spread to other places in the body and usually cannot be cured or controlled with treatment (advanced). Genetically modified therapies, such as ROR1 specific chimeric antigen receptor (CAR) T-cells, are taken from a patient's blood, modified in the laboratory so they specifically may kill cancer cells with a protein called ROR1 on their surfaces, and safely given back to the patient after conventional therapy. The "genetically modified" T-cells have genes added in the laboratory to make them recognize ROR1.