While some Iowans say they remain inspired by Mr. Carson’s life story as a person who rose from a hardscrabble childhood to become a renowned physician, faith in his presidential prospects has dimmed.

“Ben’s a good guy and he has guts, but I don’t think he has a chance to win,” said Larry Carder, an undecided voter from Des Moines. “I think out of the gate, he was just too soft-spoken. What he says is reasonable, but in your mind you’re saying, ‘Can he really lead the country?’”

Many religious Iowans who trudged out on an icy night to hear Mr. Carson said they hoped divine intervention might help him beat the odds.

“We Christians need to get on our knees and pray, and I mean, pray,” Marlene Stemler said. “My heart says, as a Christian, vote for Ben Carson.”

Changing minds of undecided voters this late in the process will be challenging.

Despite Mr. Carson’s ease talking to religious groups, endorsements from influential evangelical leaders such as Jerry Falwell Jr. of Liberty University and Tony Perkins of the Family Research Council have gone to Donald J. Trump and Senator Ted Cruz, the favorites to win Iowa.

“I was surprised by that,” Robert Dees, the retired general who is the new chairman of Mr. Carson’s campaign, said of Mr. Falwell’s backing of Mr. Trump. “But the only endorsement we plug for is ‘We the people.’”