This is an open-label, three-part, multiple-dose study to evaluate safety, tolerability, and efficacy of U3-1402 in patients with HER3-positive metastatic breast cancer. HER3 is a unique member of the human epidermal growth factor receptor, which defines a certain type of cancer. The number of patients and treatment cycles are not fixed in this study. Subjects who continue to derive clinical benefit from the study treatment in the absence of withdrawal of consent, progressive disease (PD), unacceptable toxicity, or death may continue the study treatment until the end of the trial.