Summer 1836
Col. James M. Burgess made the first claim in Lima, on Section 17; he and a partner stayed through the season, made improvements, then returned East.
Nation and worldGrade A · 1 independent source
The evidence
“The first claim made was by James M. Burgess, on Section 17, in the summer of 1836, who, with a partner, were the only persons as near as I can learn, that made any stay in the town during that season,— After staying through the season, and making some improvements, they retamed to the East and have not been residents here since that time.”
1856-history-of-rock-county/13-lima.md, PDF p. 114 / book p. 96
Guernsey, Orrin, and Josiah F. Willard, editors. History of Rock County, and Transactions of the Rock County Agricultural Society and Mechanics' Institute. Janesville, Wis.: William M. Doty & Brother, 1856. Cornell University Library via the Internet Archive, archive.org/details/cu31924028871593.
Grade A: Found on a page of one of our sources