I'm hardly technically qualified to give a full answer on this but I have a few instinctive rebuttals: i) you can transact offline in a variety of (admittedly inferior) ways, ii) it's the UTXO set that really matters, and it would be preserved just fine until a new method of global synchrony on transactions was reached, and iii) there isn't really such a single thing as "The Internet", so the question is ill-defined anyway.