The real question is whether or not world governments would allow for such a scenario. Kleptocracy is more common than one might think, and sometimes comes in tandem with plutocracy. What's unique about the most powerful government, the United States, is that it is a very different form of kleptocracy than history has ever seen. Look at every store of value/means of transacting and we see that governments have always seized the predominant technology for this at one point or another. Think the 1930s, the US gov't seized mass amount of gold and silver, also Prodrazvyorstka in the Soviet Union.

In theory BTC could exist without fiat, but then government would lose control. Unless they owned a 51% interest in mining that is, which is technically feasible given the resources available although not likely because they know what would happen in that case.

Technically speaking, BTC would be better off as an underlying store of value for a much speedier system layered on top. Kind of how gold backed the paper dollar, even though most of it was in vaults in New York/Kentucky, not in hand. BTC is digitally "heavy", just how gold is physically heavy and not easily moved en masse. The problem is, that requires trusting a third party. So someone would have to come up with a trustless way to implement that scenario. Ideally, this system would be cryptographically secure, but so speedy that the security of payment is as fast as a Visa transaction, or even better, me handing you cash.

So the answer is yes. But would it be allowed?