Inclusion Criteria:

          -  Pathologically or cytologically documented Stage IIIB/IV non-small cell lung cancer,
             or unresectable recurrent disease following locoregional treatment.

          -  For Phase 1B Extension Only:

               -  Either or both of the following: A tumor which harbors an activating Epidermal
                  Growth Factor Receptor (EGFR) - mutation; History of objective response, or
                  stable disease for at least 6 months, after treatment with erlotinib, afatinib,
                  or gefitinib.

               -  Either or both of the following: Progression or recurrence of disease after
                  receiving prior continuous gefitinib, afatinib, or erlotinib; A tumor known to
                  harbor a de novo T790M mutation, which is known to confer EGFR TKI resistance.

               -  Participants are allowed to have received systemic chemotherapy or
                  investigational therapy in the intervening period prior to trial enrollment

          -  Capable of giving written informed consent.

          -  Evaluable disease, as follows: For Phase 1A Dose Escalation: Have the presence of any
             evaluable disease, including bone metastases, effusion, or cystic metastases. For
             Phase 1B Extension Only: Have progressive and measurable disease as defined by the
             Response Evaluation Criteria in Solid Tumors (RECIST v 1.1).

          -  Reproductive potential must be either terminated (by surgery, radiation, or menopause)
             or attenuated by the use of an approved contraceptive method during and for 3 to 6
             months following the study.

          -  Participant agrees that IV bisphosphonates will be withheld during the first 8 weeks
             of dasatinib therapy due to risk of hypocalcemia.

          -  Have recovered from prior drug-related toxicity to Grade ≤ 1 Common Terminology
             Criteria for Adverse Events (CTCAE) v4, within 21 days of initiation of on-study
             treatment.

          -  Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 at initial
             enrollment, as assessed by clinician or investigator.

        Exclusion Criteria:

          -  Have previously completed or withdrawn from this study or any other study
             investigating dasatinib. Prior treatment with other tyrosine kinases, including
             afatinib, is acceptable.

          -  Prior recent systemic or investigational therapy within 21 days of initiation of study
             treatment. An exception is that epidermal growth factor receptor (EGFR) inhibitor may
             be continued up until 3 days of initiation of study treatment.

          -  Women who are pregnant or breastfeeding. Women of childbearing potential must have a
             negative pregnancy test (β-HCG test in urine or serum) prior to commencing study
             treatment.

          -  Patients with documented central nervous system or leptomeningeal metastasis (brain
             metastasis) at the time of study entry. Patients with prior brain metastasis may be
             considered if they have completed their treatment for brain metastasis and no longer
             require corticosteroids.

          -  Patients with disease progression in the central nervous system (CNS) only.

          -  Serious concomitant disorder, including active bacterial, fungal, or viral infection,
             incompatible with the study (at the discretion of the principal investigator).

          -  Uncorrected severe electrolyte disorder, including severe potassium (<3.0 mEq/L) or
             magnesium ( < 1.0 mEq/L) deficiency.

          -  Any gastrointestinal disorder with diarrhea as a major symptom, such as Crohn's, or
             pre-existing chronic diarrhea Common Toxicity Criteria (CTC) Grade ≥ 2 of any
             etiology. Included are malabsorption disorders that in the opinion of the study
             physician may affect absorption of either afatinib or dasatinib.

          -  Prior major surgery or radiation therapy within 14 days of initiation of treatment.

          -  Electrocardiogram (ECG) abnormalities indicative of arrhythmia (at the discretion of
             the investigator).

          -  History or presence of clinically relevant cardiovascular abnormalities such as
             uncontrolled hypertension, congestive heart failure New York Heart Association (NYHA)
             classification of 3, unstable angina or poorly controlled arrhythmia. Myocardial
             infarction within 6 months prior to enrollment.

          -  Baseline (< 1 month before treatment) cardiac left ventricular function with resting
             ejection fraction of less than 50% measured by multigated blood pool imaging of the
             heart (MUGA scan) or echocardiogram.

          -  Any history of clinically significant ventricular arrhythmias (such as ventricular
             tachycardia, ventricular fibrillation, congenital long QT syndrome, or Torsades de
             pointes).

          -  Prolonged QTc interval on pre-entry electrocardiogram (> 470 msec for men and >480
             msec for women per American College of Cardiology/American Heart Association [AHA/ACC]
             2011 scientific statement).

          -  History of significant bleeding disorder unrelated to cancer, including diagnosed
             congenital bleeding disorders (e.g., von Willebrand's disease).

          -  Patients currently taking drugs that are generally accepted to have a high risk of
             causing Torsades de Pointes.

          -  Patients with pre-existing interstitial lung disease (ILD), or pericardial / pleural
             effusion of grade 2 or higher. Trace pericardial or pleural effusion is acceptable.

          -  Patients who require chronic oxygen therapy for chronic obstructive pulmonary disease
             or pleural effusions (malignant or benign).

          -  Patients requiring comedication with potent P-gp inhibitors (including cyclosporin,
             azithromycin, erythromycin, ketoconazole, itraconazole, quinidine, phenobarbital salt
             with quinidine, ritonavir, valspodar, verapamil) or inducers (including rifampicin).

          -  Known active hepatitis B infection, known active hepatitis C infection, or known HIV
             carrier.

          -  Known or suspected active drug or alcohol abuse.

          -  Known hypersensitivity to afatinib, dasatinib, or the excipients of any of the trial
             drugs.

          -  Laboratory exclusion criteria: Absolute neutrophil count (ANC) < 1000 / mm^3, Platelet
             count < 100,000 / mm^3, Serum creatinine ≥1.5 times the upper normal limit or
             calculated/measured creatinine Clearance ≤60 mL/min., Total bilirubin ≥1.5 mg/dL (>26
             mol/L, SI unit equivalent), Aspartate amino transferase (AST) or Alanine amino
             transferase (ALT) ≥ 2.5 times the upper limit of normal (if related to liver
             metastases ≥ five times the upper limit of normal).