BANJA LUKA, Bosnia and Herzegovina — Milorad Dodik would like to make two things clear at the outset: that he is the president of a legitimate state, rather than an ethnically cleansed territory in a country with internationally recognized borders; and that, at 58, he can still shoot hoops like a pro.

On basketball, he has a point. He plays regularly, and although he never played professionally outside the former Yugoslavia, once an international powerhouse in the sport, he showed good game on a court in St. Petersburg last summer when he joined a veterans’ team, led by the Russian former NBA star Andrei Kirilenko, against Moscow old-timers.

He is wrong on the presidency, though. Despite his determination to break from the Muslim-Croat part of Bosnia and Herzegovina, Mr. Dodik remains the president of the Serb autonomous region, a slice of territory nearly the size of Belgium. The region likes to call itself the Republika Srpska — a name given to the blood-soaked land by Mr. Dodik’s predecessor, Radovan Karadzic, the convicted war criminal serving a 40-year sentence for genocide against Bosnia’s Muslims.

During his two terms as president of the Republika Srpska — and two more as prime minister — Mr. Dodik has ruffled more than a few feathers. Despite warnings from the international community, and in violation of a ruling from Bosnia’s highest court, he pressed for a referendum in 2016 on whether to celebrate Jan 9 as a national holiday, the day in 1992 Mr. Karadzic declared a Serb-only state in Bosnia, unleashing a genocidal war.