It has been a hallmark of the Trump White House never to admit a mistake, never to apologize and never to cede a point. This case was no different. “The White House staff will never be lectured on truth-telling from the media that pushed a flat-out lie about Donald Trump for two years,” Hogan Gidley, a White House spokesman, said in an email.

Ms. Sanders, who has often taken news media outlets to task for what they write and report and has accused them of spreading “fake news” about Mr. Trump, was just a footnote in Mr. Mueller’s 448-page report. But because of her public role, the anecdote involving her false statement loomed large in the broader portrait by the special counsel of a White House defined by a culture of dishonesty.

And it remains to be seen how the incident and Ms. Sanders’s response will affect her credibility with reporters, with whom she has had fewer interactions than many of her predecessors since the White House quietly phased out the daily press briefing over the past nine months.

More often, Ms. Sanders speaks for the president on friendly programs like “Fox & Friends.” She has also come to view her role as a person who defends her colleagues and the president, rather than someone who delivers a message to the press about the work that is underway at the White House.

Some of Mr. Trump’s aides and allies acknowledged on Friday that it was problematic for the president’s chief spokeswoman to spend airtime defending her own credibility. But White House officials — some of whom think Ms. Sanders is taking an unfair beating in the press — do not expect Mr. Trump to be fazed by the controversy. Unlike previous administrations, in which officials feared blows to their credibility in public, Mr. Trump’s press aides are generally performing for an audience of one — the president.