Yes. Some looney tunes like myself keep backups of Bitcoin on disk (HDD and CD/DVD, the latter being impervious to EMP) with the software/hardware to run them in Faraday cages. Assuming I can find a viable power source post idiot mode, it would most definitely survive. Worst case scenario it may be incomplete, but that'd be expected as consensus would have to be rebuilt anyways.
Yeah, I am sure Bitcoin will be useful after a 50 kiloton bomb is dropped on any random city. That is definitely what people will want for payment, bitcoin. Not water, food, or toilet paper... bitcoin.
A distributed system would survive if the network it was running on had enough redundant paths and bandwidth available to keep a path available after some number of attacks.