It wouldn't be a fiat currency.
fiat money: money without intrinsic value that is used as money because of government decree
(N. Gregory Mankiw (2014). Principles of Economics. p. 220. ISBN 978-1-285-16592-9)
No government decreed its people to use Bitcoin. Quite the contrary: One of Bitcoin's goals is to take control away from governments, whereas fiat currencies are a means of governments to achieve and maintain control over their peoples' funds.
You appear to mean to ask whether it's possible to distribute banknotes the value of which is backed by Bitcoin which is what I assume to be the question I'm responding to in my answer.
In principle, it's possible.
The banknotes have to be created by an entity which is:
- able to create banknotes which are hard to counterfeit and of which it's easy to spot a forged one.
- trusted to keeps its promises.
- predicted to last a long time.
The 1st point requires advanced knowledge about creating verifiable documents and the ability to keep secrets.
The second point adds the willingness to keep those secrets as well as the willingness to actually pay out money in Bitcoin if asked to and provided banknotes sufficient in value. This means that the entity has to have something to lose.
The 3rd point should be trivial. If the entity disappears after giving out the banknotes, people can't exchange them for Bitcoin.
As far as I can see, these requirements can only be met by states or large companies.
However, even though it's possible, fully backed banknotes are very unlikely to ever appear again. No matter whether they are backed by gold, Bitcoin, or anything else scarce. We live in a world where barely anything is backed. The credit a bank gives you not only isn't backed by some scarce good but also not even backed by banknotes lying in the bank's vault. Yet most people seem to be fine with this.
If people's attitudes shift to more critical thinking about money, of course, partially or fully backed banknotes become more plausible. Such banknotes can be backed by any scarce good. This scarce good very well can be Bitcoin which has great advantages over gold:
- It's easy to transfer.
- Reserves can easily be proven to exist without endangering them. And I mean actually to be proven to exist. Not just "Here is a video someone made 10 years ago, look at all those shiny gold bars we totally didn't put in there using CGI." or anything alike.
- Gold – which is a resource needed for to manufacture certain things (with far lower value than it's being currently traded for, though) – doesn't unnecessarily lie around in vaults.
- Its volume can't be inflated. There is a huge amount of gold in Earth's crust. If an easy way to mine it emerges, its available volume is greatly inflated, decreasing its value dramatically.