Depends on how you define "chainsaw", "chain saw"...
A surgical "chain saw" was made and patented and used in the 1850's-1860's.
Manually-operated continuous-loop "chain saws" were made and used in the 1880's.
Steam-powered? ...some interesting old photos exist from the 1890's-1900's...
Powered by an electric motor or internal combustion engine??
1910... 1911... 1914...
Mass-produced for the timber and logging industry?
In North America: Charles V. Wolf's saw made by Reed-Prentice in the 1920's-1930's...
In Europe: Arbor, Gerber, Sector, Tapio, made in the 19-teens...
later came Rinco... Dolmar... Stihl...
A lot of US Patents were taken in the time from 1914-1920, with detailed drawings and text descriptions.