https://en.wiktionary.org/wiki/famous#Adjective gives this counter-example:

  2. In the public eye.

    Some people are only famous within their city.

A DDG search for examples easily finds things like:

"Throughout Newberry’s long history, there has been an accumulative list of memorable and locally famous people. Machinist and inventor, Reginald S. “Reg” Ruggles was one of them.", at https://mynewberrynews.com/community/newberrys-machinist-and... :

"Most of the letters ended with a PS containing one of Aunt Agatha's aphorisms, which became famous throughout the county" - https://en.wikipedia.org/wiki/Sidney_Grapes