Trying to frame Mr. Bevin’s apparent loss as a ninth-inning rally that fell just shy of a win, Ms. McDaniel also noted that five of the other six Republicans on the ballot in Kentucky won their races and that “President Trump helped lift the entire ticket.”

Perhaps Mr. Trump’s visit to the state on Monday night did indeed help. Mr. Bevin is trailing Andy Beshear, his Democratic rival, by only a few thousand votes and is now seeking a recanvassing of absentee ballots. But public and internal Republican Party polls suggested the race had been extremely close before Mr. Trump arrived.

Ms. McDaniel’s selective presentation of the facts hews closely to the playbook of a president who sees his brand as all about winning — and who has no qualms about engaging in what he long ago coined euphemistically as “truthful hyperbole” if he thinks it helps him.

Other Republicans on Wednesday echoed Ms. McDaniel’s comment, which aligns with the effort by Mr. Trump and his allies to build up and maintain an aura of political invulnerability that does not always match reality.

It is not uncommon for supporters in either party to try to highlight their president’s political influence. But in an era where Republicans are largely defined by how fervently they support Mr. Trump, the remarks were the latest sign that any admission of weakness regarding the president or his political prospects is frowned upon.