Have you tried turning off the power supply unit for a couple of minutes, then turning it back on?
Its also possible that the power unit has failed.
If the base is dead then you wouldn't be able to reset it anyway. Although it's not clear if you've followed @Keith_Beddoe's suggestion & left it unpowered for a while. I'd suggest overnight or at least a couple of hours.
There's not much else you can do aside from replacing parts & I'm guessing the 8600 has the same proprietary connector as my 8500. So the only real option, if you don't want to replace it entirely, would be buying used parts from the likes of Ebay. You could either buy just an AC adapter, on the basis that if that doesn't fix it the base itself is dead. Or buy a complete unit, identify what's broken & resell the rest for spares.
Edit: A quick look at Ebay suggest £10 for an adaptor or £20 for a complete unit. It is indeed the same part as the 8500, MN-A003-A083.
Just ordered these for £48. The set that have just failed after about 10 years cost £64.
Should do. Only issue might be they need deregistering first, which of course would need the dead base. If that is the case, try taking the batteries out for an extended period of time. Maybe once whatever stores the registration info is exhausted they'll factory reset. But if it's written to non-volatile memory there may be a problem.
Looks like dead end. I spoke to BT technical, who didn’t think (!) that the handsets could be deregistered without the linked base unit.
There may be an alternative. On my 8500 I have a Reset option under Settings. The H/set Setting factory resets the handset, including registration. I've just tested it.