I downloaded the Instant Flash BIOS v3.00 for the board, and it successfully installed in UEFI, but the system reentered the same boot loop on restart. Sometime during this next cycle, however, Windows did start, and I was able to check that BIOS v3.00 was detected. About 15 minutes later, the system shut off by itself. It reentered the ME firmware update loop when I powered it back on. It does still detect the problem and allow access to UEFI after about ten cycles, but I cannot bypass the loops using the keyboard - the ASRock splash screen that shows the keyboard boot options appears immediately before the ME update, but it does not respond to keyboard input.

My guess is that the Intel ME firmware is corrupted, and the Instant Flash BIOS update isn't touching that region of BIOS. The most promising information I've found so far describes advanced BIOS management using the FTK tool (example:  http://www.overclockers.com/forums/showthread.php/728348-How-to-fix-your-BIOS-if-there-is-corrupted-ME-area" rel="nofollow - http://www.overclockers.com/forums/showthread.php/728348-How-to-fix-your-BIOS-if-there-is-corrupted-ME-area ), but I can't tell if any technique based on this would work with this mobo.  I'm still waiting for a response from ASRock tech support, but given the age of the mobo, I wouldn't be surprised if I don't get any help that way.

The details are on ASRock's support page under Repair/RMA. 

If you do go this route you can always try the method you found first and see if it sorts the issue. ME corruption can be a real pain. Fairly often we see users having to order a new BIOS chip or resorting to using an external ROM flasher to force flash the chip completely rather than a partial update. 

Let us know how it goes 

Is it accurate to guess from your reply that ASRock BIOS is not commonly user-managed or modified, in contrast to ASUS or Gigabyte? These are the two mobo manufacturers I have seen referenced in the forum posts regarding FTK and other BIOS management techniques. I was hesitant to try any of those since a) I haven't found any posts describing use of these tools on ASRock mobos, and b) ASRock doesn't provide BIOS files in similar formats to those manufacturers, making it harder to use the tools.

Any idea how long it typically takes to receive a new BIOS chip from ASRock? I see several suppliers on eBay that say they'll pre-configure a chip once you order it and get it to you within a week.