Another analyst, Marian Lesko, said in an interview on Sunday: “Compared to four years ago, Slovaks have realized the threat of extremism a lot more and they’ve expressed it.” Mr. Lesko also said he believed that voters had grown tired of the Smer party.

Image Jan Lunter, the new governor, in Banska Bystrica, Slovakia, last month. “I’ll open all the doors and windows and let in some fresh air,” he said on election night. Credit... Martin Mikuta/Czech News Agency, via Associated Press

Mr. Kotleba was first elected governor of Banska Bystrica, in central Slovakia, in 2013 and became a member of Slovakia’s national Parliament three years later, along with 13 other members of his party, known as Kotleba — People’s Party Our Slovakia. His party also faced a retreat in regional Parliaments, winning just two of more than 400 seats, one in Nitra and the other being a seat that Mr. Kotleba himself retained. (Mt. Kotleba remains a member of the national Parliament.)

He lost the governorship on Saturday to a businessman, Jan Lunter, whose candidacy had been supported by the governing coalition and the opposition.

“First thing I will do in the governor’s office will be that I’ll open all the doors and windows and let in some fresh air,” Mr. Lunter said Saturday night. “And I think it’s important that we go back to European Union again, we will put up its flag in the office again, back where it belongs.”

“We belong in Europe,” he added. “We don’t want to leave.”

President Andrej Kiska, an independent, said after casting his ballot on Saturday that it was time for voters to decide “which direction our country is going to take, for the extremists, the neo-fascists, to not take their place.”