What is the country with the smallest population to win an Olympic medal?

The Most Serene Republic of San Marino, with a population just north of the 30k mark. Previously, this record had been held by Bermuda (pop. c. 64k).

San Marino has participated in the Olympic games for some 61 years (since Rome in 1960). Trap shooter Alessandra Perilla picked up the bronze medal in the Women's event at Tokyo (2020/2021). A couple of days later, Perilli and Gian Marco won silver in the new mixed-gender Trap event.

https://www.npr.org/sections/tokyo-olympics-live-updates/2021/07/29/1022160238/tiny-san-marino-population-34-000-just-won-its-first-ever-olympic-medal
https://en.wikipedia.org/wiki/San_Marino_at_the_Olympics