Hyperactive trackpad

All of a sudden my trackpad (with an iMac, wireless trackpad and keyboard) seems to have overdosed on amphetamines. Even when I don't touch the trackpad I get frequent clicks and occasionally pointer movements. Sometimes ten or more a minute, enough to make it difficult or impossible to work.

I've tried what I could think of:

Made VERY sure I wasn't inadvertently touching the trackpad. Nope.

Rebooted. Nope.

Went to System Prefs and turned off both single-click by tap and secondary click with two fingers. Nope.

Replaced the batteries (they were down to 30%). Nope.

Got paranoid -- turned off screen sharing and turned on the firewall, in case someone was taking over the computer. Nope.

At this point I'm pretty much down to (a) hardware malfunction on the trackpad, (b) electrical interference somewhere, (c) weird software bug on the iMac, (d) hardware malfunction on the iMac.

I'm pretty sure it's not hardware on the iMac, because removing the batteries from the trackpad cures the problem. And for the same reason I'm almost certain that it's not software on the iMac that's doing it.

Next step, I suppose, is to borrow someone else's trackpad and see if that solves the problem. Any other ideas?

-Lofty