No. Persistence of the data does not require continuous power. But the black box has a battery-powered "pinger" that produces sound to make it easier to find. The batteries for that is only required to last for 30 days.
Airliners also carry emergency locator transmitters† (ELTs), which, when activated by a crash, transmit radio signals which can be triangulated via satellite. However, ELTs do not work underwater. No ELT signals from MH370 were heard.