Hadn't thought of this.... but DEET will mess up many types of plastic resin, which is what many wood finishes are nowadays.
As long as you're not wetting the surface with the stuff, though, I'm not sure how much of a problem it would be. I'm a canoeist, so I'm always using bug dope and then handling wood canoe paddles, and I've never noticed any effect on the finish on those. But I don't use sprays -- just 100% DEET out of one of those little bottles; I put 2-3 drops on my hands, then carefully apply that around my hairline and neck, wipe off what's left around my ankles or wherever, then wipe off my hands on a cotton hat or bandanna... that's enough for anywhere I've ever traveled, including the Arctic, so there's really no need to use more.
I've never noticed any effect on anything I've touched after that. But I have messed up a couple of cheap watches during the initial application...
And if I were fortunate enough to have a Perazzi, I think I'd remember to wipe it off at the end of the day!