As the power elite of Washington filed out of the White House Correspondents’ Association dinner on Saturday, chattering about the rough-and-tumble set by the comedian Michelle Wolf, the White House press secretary, Sarah Huckabee Sanders, was invited to pose for a portrait with her fellow honorees on the dais.

Ms. Wolf smiled for the camera. But Ms. Sanders declined, not wanting to appear in a photo alongside the comedian, according to two people who described the exchange.

It was an early ripple of the tsunami to come. Ms. Wolf’s punch lines about Ms. Sanders and Kellyanne Conway, another prominent woman in the Trump administration, set off a high-dudgeon debate that raised the possibility that Saturday’s edition of the nearly 100-year-old dinner could be the last of its kind.

“It will not surprise you to hear that my email inbox is overflowing with advice on how to improve the dinner,” Olivier Knox, the incoming president of the White House Correspondents’ Association, said in an interview on Monday. He reeled off the possibilities now on the table: “No entertainer. No comic. A serious speaker. Maybe a musician. Maybe don’t televise it.”