How to pull siding nails without damaging log siding

I have a building with expensive log siding. I need to remove several rows of siding, and reinstall them later. The siding is attached to the studs with siding nails. The nail heads are either flush or slightly below the surface.

Normally I would use a cats paw, but doing so here will gouge the wood--either slightly or pretty bad. I've been wracking my brain trying to come up with a way to get those nails out without damaging the wood. Prying the pieces out is not an option. This is tongue-and-groove siding.

I'm hoping someone out there has a clever trick for this...