*Update*
Switching the battery switch off then on caused it to power on properly. Sorry for the disturbance!
On Monday night, as I turned my Gigabeat F40 off, I got an error message:
*PANIC*
Event Not Found
I hit the power button again and my device did shut off, but now I can't turn it on. Its remaining battery power was over 90%, and it can still be charged, so I know that's not the problem. Its Rockbox revision number is 17111-080415.