Danny Vowell / AP file

Bruce Jenkins (from left), Mike Sandbrink and Kate Conkey battle for control of a basketball during a charity donkey-ball game on Jan. 21, 2005 in Pembroke, Ky.

“I think it is probably somewhat confusing to the donkey,” Heleski said, but no more than pony rides or novice horseback-riding lessons are to those animals. Unlike horses, she said, donkeys tend to be calm and stoic. “And if they get confused,” she said, “they just plain stop.”

Although donkey ball players are required to sign waivers before they ride and some companies say they hold liability insurance, participants have sued after being injured. In 1997, a federal jury awarded an Illinois man $110,000 after a donkey kicked him in the knee. A fifth-grade teacher in Florida settled a lawsuit last year with the Diocese of St. Petersburg after falling off a donkey in a game in 2003.

“It sounds funny and laughable, but people get hurt,” said Louis Kwall, the teacher’s lawyer. “The real issue is the unpredictable nature of dealing with an animal.”

Donkey ball operators say that the claims are exaggerated and that their animals are well cared for. At the game here, Sisters dug his fingers into the thick coat of Timebomb, who answers to Cuddles when she’s not on the court, and gestured to the other donkeys standing nearby. “Feel them,” he said. “They’re spotless.”

His wife, Letitia, said the animal-welfare groups had never called or visited to learn about the company’s operations. “We invited PETA here,” she said. “I tried to contact them, and not once has anybody visited our farm. Normally when you see animals, people are in control. In this forum, the reason it’s so entertaining to children is that the donkeys are totally in control of the situation.”

Phelps said that PETA had never received an invitation to visit a donkey ball business but that the organization regularly received complaints from spectators. To some who watched the game here, the event seemed to evoke another era, before liability waivers and animal-welfare protests. Gina Busiak, who attended with her daughter and son, said she watched donkey ball games when she was in high school, in the 1970s.

“I was surprised they were doing it,” she said. “I thought it was a thing from the ’70s. There’s a lot of things from the ’70s that we don’t do anymore.”

This article, “Donkey Ball Stubbornly Holds On Despite Criticism,” first appeared in The New York Times.