During Phase I, the eribulin dose will be assigned upon study enrollment. The maximum tolerated dose (MTD) determined in Phase I will be the dose used for Phase II. Eribulin will be administered intravenously (IV) over 2 to 5 minutes on Days 1 and 8 of each cycle. Carboplatin area under the curve (AUC) 6 will be administered IV over 15 to 30 minutes on Day 1 of each cycle. Trastuzumab will be administered as an IV infusion on Day 1 of each cycle. A loading dose of 8 mg/kg of trastuzumab will be administered over 90 minutes on Cycle 1 Day 1. Then, a maintenance dose of 6 mg/kg of trastuzumab will be administered over 30 to 90 minutes on Day 1 of Cycles 2 - 6.