Depends how you define fiat. "State created unbacked money." is very narrow. If you take it more generally it's word magic giving things false value. Which means it's as old as a snake in the garden. 🍎

Fiat means declaration or consent. Like signing a deal. 🐍

Usury is a word magic trick that claims to make money time travel in exchange for giving someone a claim over you. 

Once people start doing that, the borrower is the servant of the lender and centralization begins. Then make the debters enforce on each other. This gets bigger, enslave dukes, then princes, then kings. Now the "rulers" and slaves to debt, enforcing debts on their people and on each other.  This is the state. And war becomes just another way for bankers to enforce and extract.

By the time you get to large centralized governments issuing fake money, hundreds of years of word magic corruption have already happened.