There are 1 US Post Offices in Clark County serving a total of 35,262 residents equating to 35,262 residents per post office. It's estimated that approximately 56,282 packages are handled across the 1 county post offices annually. Clark County is a county located in the U.S. state of Kentucky. As of the 2010 census, the population was 35,613. Its county seat is Winchester. The county was created in 1792 from Bourbon and Fayette counties and is named for Revolutionary War hero George Rogers Clark. Clark County is part of the Lexington\u2013Fayette, KY Metropolitan Statistical Area."