President Trump was in an especially ornery mood after staff members gently suggested he refrain from injecting politics into day-to-day issues of governing after last month’s raucous rally in Arizona, and he responded by lashing out at the most senior aide in his presence.

It happened to be his new chief of staff, John F. Kelly.

Mr. Kelly, the former Marine general brought in five weeks ago as the successor to Reince Priebus, reacted calmly, but he later told other White House staff members that he had never been spoken to like that during 35 years of serving his country. In the future, he said, he would not abide such treatment, according to three people familiar with the exchange.