“I’m glad I had my boots on tonight,” Mr. Perry said, “because I sure stepped in it out there.”

Supporters of Mr. Perry groaned, with one contributor saying by e-mail: “It’s over, isn’t it?” One of his rivals, Representative Michele Bachmann of Minnesota, offered pity, declaring: “We all feel very badly for him.” And Republican operatives almost uniformly declared it as a sign of great trouble for his candidacy, with Mark McKinnon, an aide to former President George W. Bush, describing the moment as the “human equivalent of shuttle Challenger.”

“I think the biggest question now is whether or not he can raise any more real money,” said Mike Murphy, a Republican strategist. “A donor strike will totally cripple what’s left of his campaign.”

It remains an open question whether Mr. Perry will be able to move beyond the moment, particularly given that the video was already looping around the Internet and television broadcasts. But his path to the presidential nomination grew more difficult, and his rivals began furiously working to present themselves as the best alternative to Mitt Romney.

The lapse by Mr. Perry was the most memorable of a two-hour debate on CNBC that was otherwise dominated by polite exchanges over economic policy. It was not until several minutes later, when he received another turn, that he explained himself, saying: “By the way, that was the Department of Energy I was reaching for a while ago.”

It was clearly a blow to Mr. Perry just as he was investing heavily in reintroducing himself to voters in Iowa and New Hampshire through television advertising. It could be weeks before it is clear whether it has a permanent effect on his campaign, but at a minimum it left him facing additional questions about his candidacy.

“It was a political death knell,” said Sara Taylor Fagen, a Republican strategist who advised Mr. Bush. “There’s just no recovering from a moment like that when you’ve had such a bad record of debates.”