For a long time, Ben Carson’s campaign team feared that his habit of inflammatory remarks would sink his presidential hopes. They sent him to media training in Texas. The candidate pledged to police his words.

But ever since Mr. Carson said on Sept. 20 that he did not think a Muslim should be president, then refused to retract the statement amid a furious reaction, his campaign has watched grass-roots support grow and donations pour in — and advisers have backtracked, deciding, in the words of one, to “let Carson be Carson.”

Mr. Carson, a retired neurosurgeon seeking the Republican nomination, has seemed lately to be a candidate unbound. He has uttered remarks on policy and national events, some divisive and some seemingly uninformed, that have led commentators on the right as well as on the left to question his fitness for the presidency.

And yet none of this has deterred elements of the Republican base, which in making 2015 the year of the political outsider see in Mr. Carson’s provocative comments a more palatable variation on the bombastic insults of Donald J. Trump. Both accuse critics of “political correctness.” Now Mr. Carson is edging up behind Mr. Trump in many polls, with Carly Fiorina, a third outsider candidate, close behind.