Kodak easyshare zd710 battery issue

I've got a similar issue with my Z712IS. When it was new, I could change the battery and the camera would retain the clock for a minute or two while changing it.

Today, even if I rapidly swap the battery in 1-2 seconds, the camera will lose its time/date settings.

The official word is that this shouldn't happen - that the camera has capacitors that will retain the time for a short time while the battery is removed. If that's so, then the caps have died, but I have no clue how/if they can be replaced.

I think it is more likely that there is an internal battery (or maybe a supercap) designed to run the clock independent of the battery pack, and it needs to be replaced. But without access to any service manual to teardown document, I can't prove that.

Update: I found the teardown for the Z650, which shows what appears to be an internal battery: Kodak Easyshare z650 Teardown. So maybe there is one in a Z712IS or a Z710 as well.