We present a new millimeter-scale bimorph actuator for microrobotic applications, driven by feedforward controlled shape-memory alloy (SMA) wires. The device weighs 10 mg, measures 14 mm in length, and occupies a volume of 4.8 mm3, which makes it the lightest and smallest fully functional SMA-based bimorph actuator for microrobotics developed to date. The experimentally measured operational bandwidth is on the order of 20 Hz, and the unimorph and bimorph maximum low-frequency displacement outputs are on the order of 3.5 and 7 mm, respectively. To test and demonstrate the functionality and suitability of the actuator for microrobotics, we developed the Fish-&-Ribbon-Inspired Small Swimming Harmonic roBot (FRISSHBot). Loosely inspired by carangiformes, the FRISSHBot leverages fluid-structure interaction (FSI) phenomena to propel itself forward, weighs 30 mg, measures 34 mm in length, operates at frequencies of up to 4 Hz, and swims at speeds of up to 3.06 mm/s (0.09 Bl/s). This robot is the lightest and smallest swimmer with onboard actuation developed to date.