Yes it can exist but the problem is that it will have the same role as fiat meaning that it will become stable and somewhat centralized. Plus, bitcoin wouldn't be an asset anymore because it finally turned to be a currency.
It's like imagining life without eating. Bitcoin as a new generation currency that is still not recognized by many countries in the world. Fiat currency with a fixed exchange rate and has been recognized as valid as payment. Bitcoin without fiat will still work but fiat is the main currency that is recognized as legal by the world. So fiat is legal for living expenses paying bitcoin is a consensus that has been agreed worth can be used as fiat. So they should stay in touch with each other for now. But the future may change if BTC has become a legal tender and is recognized worldwide.