I have two issues. As long as the camera battery is charged, the Date and Time will stay updated. My issue is a fully charged camera battery (I have 3) will die within 24 hours of placing it in my camera. My camera will not start up. When I replace the battery with another one fully charged, the Date and Time is lost. No two weeks of remembering the Date and Time as mentioned by Canon. My camera is usable after inserting a charged camera, resetting the Date and Time, then using the camera as needed. When I turn the camera off for a while, then turn it back on, the camera battery shows as plenty of power. If I try it the next morning, it is dead. This happens even when the auto Date and Time with GPS is disabled.
It sounds like the built in rechargeable battery for the date and time in your camera is shot. That battery is not user repalceable as far as I can tell. Have you tried calling Canon to see what they would charge to replace that battery although it might be built in to one of the circuit boards. Which could mean a pricey repair.
How old is your camera ?