https://en.wikipedia.org/wiki/Firefighting_in_ancient_Rome

And private firefighters were active even earlier, with Crassus particularly notorious:

https://en.wikipedia.org/wiki/Marcus_Licinius_Crassus

Fully paid, professional municipal departments came in 1853 Cincinnati.

So I guess it depends on how you define it. US gets a lot of credit for the idea of a “fire department”