Matador Resources Co. is suing U.S. immigration authorities to give the oil driller’s $110,00-a-year big-game hunting expert a visa.

Roy Dirk Ludick, a professional hunter and guide licensed in Zimbabwe since 2003, plays a critical role in Matador’s outreach to “high-valued partners, shareholders, and stakeholders,” the company said in a lawsuit against the U.S. Citizenship and Immigration Services.

The agency erred in denying Matador’s request to classify Ludick as an alien “of extraordinary ability in business,” according to the suit. Matador said Ludick is so good at what he does that he makes more than twice what other guides rake...