With Mr. Carson suddenly surpassing him in national polls, Mr. Trump’s instinct to attack, with force and repetition, was on full display during a series of television interviews. It was an intriguing turn for Mr. Trump, given his long history of shading the truth and later admitting it, sometimes under oath.

In preparation for a lawsuit he filed claiming defamation, Mr. Trump was asked during a 2007 deposition if he exaggerated about the value of his real estate holdings. “I think everyone does,” Mr. Trump replied. Pressed further, he said he exaggerated “not beyond reason.”

In a precursor to Mr. Carson’s woes, Mr. Trump has acknowledged embellishing a claim about himself in two books he wrote. As he described his financial comeback, Mr. Trump claimed that his debts had reached $9 billion in the 1990s. But in the 2007 deposition, Mr. Trump conceded that the figure was incorrect. (“That is a mistake,” he said, “and I don’t know how it got there.”)

Such fact-bending has extended to his campaign for president. During the last Republican presidential debate, Mr. Trump denied that he had called Senator Marco Rubio of Florida the “personal senator” of Facebook’s chief executive, Mark Zuckerberg, an advocate of immigration reform. Mr. Trump’s own campaign website had, in fact, described Mr. Rubio that way.