I would like to comment on the various "fixes" suggested by other readers, and the various presumed causes.
First off, my tests would suggest that at least in most cases the wireless environment is NOT the cause of the machine going to sleep -- even pushing the touch-panel and/or the power-on/off button has no effect. That's just WRONG!
Secondly, other wireless devices and addresses maintain their operability even when the Officejet 8500 A910 has fallen over, thereby proving that the problem lies with the printer and not the network/router configuration.
Thirdly, simply powering the printer down through the simple expedient of unplugging it and plugging it back in will (after the several minute restart sequence) return the printer to a viable state. Admittedly, this isn't something that can be done remotely unless you have a network-access powerstrip driving the printer (and they are certainly available).
What apparently has happened is that the printer has lapsed into a sleep state whereby all normal interrupts have been masked out completely except for the power-on-pulse (POP) that coincides with a cold restart. As an electronics engineer who has designed equipment, I'm fairly confident of my conclusion.
This represents an intrinsic firmware problem in the printer itself.
Admittedly I haven't tried plugging it into a USB cable and driving it that way (as several other correspondents did) when the system has fallen over; enough other functions have been compromised (touch-panel/power-on switch/etc.) that I would not find such an artificial solution satisfactory. A firmware bug is a firmware bug is a firmware bug; until it is fixed (and tested rigorously) artificial bandaids are merely that and nothing more.
Anyone at the HP Help Desk who tells you the problem is due exclusively to a network issue is probably reading from a script which in turn was devised by someone not intimate with the state-table for the printer.
Sadly, this is symptomatic of HP's continuing reputation -- they make great (or at least very, very good) hardware, but their software/firmware/tech-support is perhaps less robust.