Further searching the internet made clear to me that there is a structural problem with the backup capacitor of the m100. Further details, including replacement part stock numbers, I found on the URL below:
Q: How do I change my Palm m100's AAA batteries safely?
A: According to the Palm documentation, you have one minute to change
batteries, which should be easy if you the new batteries unwrapped,
correctly oriented, and within easy reach. Make sure you put the
batteries in with the correct orientation, one + terminal in each
direction. Many Palm units will hold data for a lot longer than
1 minute, but don't count on it.
If you cannot change batteries at all without losing data
(repeatably; absolutely under 1 minute; and you've made sure the
batteries are actually good by testing them with a voltmeter), you
could have a damaged Palm (might be a dead backup capacitor, maybe a
0.10F 5.5v SGtype; maybe Panasonic EECS0HD104H or Digikey part
number P6952-ND or P6972-ND, depending on which model of Palm/Pilot).