although heavily misunderstood, this is built into cryptocurrencies since day 1 (many critics have long thought crypto requires power and internet access, many proponents also don't know otherwise)
with card networks learning from competition and functionally being public-only keys, this should be even simpler to implement
Checking the signature on some blob that says "this be money" is not enough.
For example, when each transaction is done, both parties might keep a cryptographic proof which they are required to submit once they are online again.
Failing to submit could result in a small fine (to encourage submission) and double spending which can then be detected could result in a large fine (or even a prison sentence), for example.
There is, perhaps, a privacy issue, just like with blockchain. But it's not more of an issue than online transactions.
The offline credit card system does not proof fraud but just has insurance.
Offline payment specifically allows double-spending. If you have offline payment, you don't need a ledger at all; you can just use PGP.
Even better, when you're back online the network will run a sanity check to protect against the evils of inflation.
(Note: inflation-check fees will vary between 0 and 100% per payee.)