WHITESBURG, Ky. — Moonshine packs a punch in this corner of Appalachia, where making hooch is steeped in local lore. But when Colin Fultz, the grandson of a bootlegger, opened a gourmet distillery here last fall, he ran afoul of a spirit even more potent than white lightning: University of Kentucky basketball.

With his outlaw grandfather — who spent 18 years behind bars for smuggling — very much on his mind, Mr. Fultz, a businessman and onetime coal miner, set out to carry on his family’s tradition in a legal, and thoroughly modern, way.

He tinkered with recipes, blending peaches and blackberries into mash brewed in his garage. He hired a lawyer — “My wife got on me, said I was going to get into trouble,” he said — and renovated an old car dealership, where he now distills and sells fruit-infused whiskey, serving it in thimble-size cups from an exposed-brick tasting bar.

But Mr. Fultz also tried to trademark his business name: Kentucky Mist Moonshine. And that, sports lovers, is how a moonshine maker wound up suing the University of Kentucky — the basketball behemoth exalted by its “Big Blue Nation” of fans — in federal court over a fundamental question: Who owns the rights to the name of the state?