Cue flame war and enraged comments that usually follow comments of this type. Eventhough saying that bitcoin does not require the internet to survive sounds ludicrous, it is not as unreasonable as it sounds at first glance.
In our glossary, bitcoin simultaneously means multiple things at the same time.
Bitcoin is the concept of cryptographically signed transactions stored in a global public ledger. This ledger is a chain of blocks that rely on the proof of work algorithm for enforcement of their authority.
Bitcoin is the current blockchain as it exists. The current history of transactions as exists on BlockExplorer. Taking the current blockchain, I could present it as bitcoin. BlockExplorer does not add information to the blockchain, but we do say it is a participant in bitcoin.
Bitcoin is the active functioning system. This includes wallet services who interact with the blockchain and create new transactions. It includes people sending funds to each other and making trades for fiat currency on exchanges where the bitcoins never touch the blockchain. It includes people who make offline bitcoin transactions and trade them between each other.
Bitcoin is the peer to peer protocol that runs over the internet. For the non-techies, a protocol is the language that computers speak between themselves. Bitcoin nodes on the peer to peer network, have a commonly agreed language that they speak between themselves. This language, the bitcoin protocol, is called bitcoin colloquially among developers and is the most correct meaning for the term bitcoin from the perspective of a developer.
Bitcoin is the original Satoshi client, participating in the bitcoin system together with BitcoinJava, libbitcoin and BitcoinJS. There are graphical interfaces built on top of these core software bases such as bitcoin-qt, Electrum, MultiBit, Spesmilo and Safebit.
It is unfortunate that the term bitcoin is ambiguous. Ambiguity leads to arguments when it is not clear what one is discussing. I once had a passionate argument with an American friend of mine, when eventually we discovered that the British billion (1012) is different to the American billion (109).
Viewed in this way, we realise that for the bitcoin system to function, we can have multiple competing protocols and software implementations of the protocol. It is entirely possible to create new protocols to deliver bitcoin traffic over email, over telephones and by carrier pigeon.
The nice thing is that bitcoin can exist in other internets and mesh networks, or on highly anonymous darknets. There is nothing to stop the entire global system functioning as long as there are gateways between each protocol.
Bitcoin traffic is low too. Blocks are not particularly large in size, making it feasible to envision the blockchain being distributed in all manner of ways. Miners do not need to be constantly attached as long as they are notified of new blocks in a not-so-immediate way every 10 minutes. Normal participants can get away with even more.
This is what people mean when they say bitcoin can function without the internet. And it is not entirely incorrect.