DS-1205c With Gefitinib for Metastatic or Unresectable Epidermal Growth Factor Receptor (EGFR)-Mutant Non-Small Cell Lung Cancer

This study has two parts: dose escalation and dose expansion. The primary objectives are: * For Dose Escalation, to assess the safety and tolerability of DS-1205c when combined with gefitinib in the study population and to determine the recommended dose for expansion of DS-1205c when combined with gefitinib in the study population * For Dose Expansion, to assess the safety and tolerability of DS-1205c when combined with gefitinib in the study population. In Dose Escalation, after a 7-day run in period (Cycle 0), there will be 21-day cycles (Cycle 1 onward). In Dose Expansion, there will be 21-day cycles. The number of treatment cycles is not fixed in this study. Participants will continue study treatment for 36 months unless they decide not to (withdraw consent), their disease gets worse \[progressive disease (PD)\], or side effects become unacceptable (unacceptable toxicity).