The ability to actively process transactions requires a sufficient number of mining computers linked via low latency connections, but as long as there's enough in operation somewhere on the planet, the blockchain will stay alive until the damaged internet infrastructure can be repaired and rejoin the network. Even if everything went dark across the globe, there's a complete history of every prior transaction and balance status of each bitcoin address that is permanently recorded and synchronized on the copy of the blockchain of every participating active node at the time of the catastrophe. With this record intact, even after a period of dormancy to rebuild, the network could restart where it left off.
There must be a fair number of EMP-hardened nodes spread around the world that would preserve the blockchain until it could be brought back online, if there's sufficient will and resources available after such an extinction level event. Most EMP events, short of the ELE variety, will not result in a global blackout, so having the ability to regenerate one's keys after the infrastructure comes back will prevent a total loss of funds. The ability to relay transactions wirelessly has been demonstrated, via amateur radio, as well as mesh networks like the Gotenna. I'm not sure how the pen and paper transactions work, but might be good thing to learn.
