From when I pulled one of the P38 nav units apart many years ago (probably 2012 when I got my P38), I don't remember it having a battery in it.
I think from memory, and the image I found of the top side of the PCB that it probably stores it's last location data in EEPROM or flash memory when it powers down. Mine was always thinking it was somewhere else, and would always go back to that location on next power up. I didn't really look too far into it, but I think it is probably the EEPROM or flash memory chip failing and not writing new data to it.
I went with the aftermarket nav route, so didn't dig further into where the EEPROM is on the board, or if it's replaceable, or if it includes anything else in it's memory - firmware or the likes.
I seem to recall a repair service for them somewhere in eastern Europe, but don't know if that is still happening!