I don't know about specifics of availability... but sure, there should be battery powered continuous lights.

Basic principle: Continuous light (sunlight, incandescent, or LED) has no sync speed limit... Continuous light all works as "High Speed Sync" at any shutter speed. That is what HSS is, is that the speedlight is converted to flash very rapidly (tens of thousands of times per second) to simulate continuous light. Then there is no sync concern, any shutter speed works (as in daylight for example). High Speed Sync rapid flashing is technically a big deal to create, but the principle is simply a name, continuous light has no sync concerns. Simply because that light is continuous. Any shutter speed works, including high speed shutters.

Fast shutter speeds do drastically reduce the continuous exposure (same as daylight does), but opening the aperture to Equivalent Exposures compensates for the shutter speed falloff. Because it is continuous.

Fluorescent is possibly an exception, continuous except older magnetic ballast lights flicker slowly (with AC line voltage cycles), and higher speed shutters won't work well. We are advised to exactly match the shutter speed to the AC voltage frequency to avoid seeing random intensity and color variations.

However CFL bulbs and other electronic ballast flickers extremely fast (tens of thousands of times per second), and so does work at higher shutter speeds.