Aside from a microSD card/recording problem, rebooting over and over again does sound like insufficient power getting to the camera. When the camera starts to record, it demands more power than startup, idle (no card installed) or the menu mode and if that power is not available due to resistive losses in the power circuit or whatever, the input voltage may drop off to the degree that the A119S senses that power has been pulled and turns itself off. Then with the load removed, the input voltage again rises and the A119S senses that power has been restored and turns itself on and tries to boot up again..repeating on and on.
1. You probably have already done this, but if you are connecting power thru the GPS mount, switch to the camera side input. GPS, if the mount is attached, should still work. There can be excessive resistive losses powering through the GPS module which can cause the power cycling problem.
2. Unlikely since you have already tried alternate power sources, but USB cables and +5Vdc power supplies are always suspect for the A119S, but try different combinations if you have that option. First try the original cables and Cig adapter (12 to 5Vdc converter) that were supplied with the A119S.
3. Does the REC button red light stay on a few (typical is 5-7) seconds after power is pulled or just does it turn off immediately. If the latter, the problem might be bad capacitors, though your symptoms are not typical for this problem. There are videos on disassembly this forum and YouTube, but opening the camera body might void your warranty so contact your seller first. If you do look inside and the two large electrolytic capacitors are swollen, then they may need replacement User replacement is possible if you have some soldering skills and the need.
Just a note, that sometimes corrupted files can be recovered after being repaired by first playing them back in the camera.