Believe it or not, it's possible. While desperately searching for solutions for my months-dead Axim, I stumbled upon this site:

http://tom61.arcadecontrols.com/axim_bootloader.html

The Axim has a bootloader, just as the iPaq does. It'll load a ROM from a CF card. I tried this on my months-dead Axim and it fixed it right up.

That rocks! Most of us know about this already though. If you see a "dead" Axim on eBay going for the same price as working Axims, no doubt the bidder is in on the secret too

Oh, ok, sorry. Nobody knew when I asked about recovery three months ago, so I figured it wasn't well-known. My bad.

No problem! Thanks for sharing.