How old is the unit?
Before you send it off to Garmin's Flat Fee Repair Service, Be sure the unit is getting voltage when you plug it in.
Attempt to charge for 2-3 hours using the USB charger plugged into your running computer.
While it's charging - inspect your car cable. The tip that plugs in the vehicle has a removable tip. Unscrew it and make sure the fuse is good.
After verifying the cables are getting the power to the unit - and it is still behaving similarly - it is safe to say the battery has failed.
CLICK HERE for Garmin's FLAT FEE repair service for out-of-warranty GPS.
Yours is listed as a $79 repair.