CHARLESTON, S.C. — At least three times this month, Joseph R. Biden Jr. has asserted that he was arrested as he sought to visit the anti-apartheid icon Nelson Mandela in prison, even saying that Mr. Mandela later thanked him for going to such an effort.

And for a week, Mr. Biden’s campaign declined to answer questions seeking comment and clarification on those remarks, which were rebutted by a former United States ambassador to the United Nations in an article in The New York Times. Mr. Biden did not mention the episode in his memoir, had not spoken of it prominently on the campaign trail, and a review of available news accounts by The Times did not turn up any mention of an arrest.

But on Tuesday, Kate Bedingfield, a deputy campaign manager, said Mr. Biden was referring to an episode in which he was separated from black colleagues in Johannesburg while on a congressional delegation trip to South Africa in the 1970s. It was the campaign’s first explanation to date — but one that still left many questions unanswered and did not square with Mr. Biden’s most recent remarks.

“He was separated from his party at the airport,” Ms. Bedingfield said when pressed by reporters following Tuesday’s presidential debate here.