Updated at 9 a.m. E.D.T.

President Trump defeated Ambassdor Nikki Haley in the New Hampshire Republican primary Tuesday night, calling into question whether the former South Carolina governor has a path to victory for the Republican nomination. In the Democratic primary, President Biden was projected to have won via a write-in campaign.

The Republican race was called by the Associated Press at 8 p.m. when polls closed. As of this morning, with 91 percent of the votes counted, Mr. Trump was leading Mrs. Haley with 54.5 percent of the vote to her 43.2 percent, the AP reports. Mrs. Haley, though, is vowing to stay in the race.

“New Hampshire is first in the Nation. It is not last in the nation,” Mrs. Haley said in her concession speech Tuesday night. “This race is far from over. There are dozens of states left to go.” 

Mr. Trump’s win by some 11 percentage points in the Granite State, where independents can vote in the Republican primary, though, casts doubt on whether Mrs. Haley has any viable path to the nomination. The contest next moves to South Carolina — Mrs. Haley’s home state — where President Trump is polling 30 points ahead of her.

The primary map after that also favors Mr. Trump. The next open primary, where independents can participate, is Michigan, but Mrs. Haley is polling at only 15 percent of Republicans there.

Mrs. Haley’s campaign had tempered expectations of a win in recent days. In a memo released early Tuesday, Mrs. Haley’s campaign manager, Betsy Ankney, said New Hampshire is not do-or-die for the campaign. “The political class and the media want to give Donald Trump a coronation. They say the race is over,” the memo says. “We aren’t going anywhere.”

Recent polls predicted Mr. Trump’s win in the Granite State, but Mrs. Haley was banking on a strong showing from New Hampshire’s undeclared voters — who make up 40 percent of the electorate — to give her an upset. New Hampshire has a history of delivering upsets, and undeclared voters can cast a ballot in either party’s primary. Mrs. Haley courted these voters, as well as anti-Trump Republicans.

“I feel a vote for Nikki Haley is a vote against Trump, so that’s why I did it,” an undeclared voter and retired occupational therapist, Jan Fraser, told the Sun at a New Hampshire polling site. “Anything I can do to prevent that insane mad man from getting into office is what I want to do.”

Yet Mr. Trump’s victory Tuesday shows his hold on the party is still strong— stronger than many politicos expected. Mr. Trump won Iowa last week with 51 percent of the vote. In New Hampshire, a more moderate state, and with the field winnowed down to two, Mr. Trump only increased his vote percentage.

“There was always a notion from the Never Trumpers that if you could get this down to a one-on-one race between Trump and somebody else, he’d be finished,” a New Hampshire Republican strategist, Matthew Bartlett, told the Sun. “And now it appears, quite the opposite,”

“The GOP is Maga Land,” Mr. Bartlett said.

The Sun spoke with several Trump voters at Portsmouth polling locations — one of the most liberal parts of the state — who said immigration, foreign policy, and the economy were their top concerns. Some were undeclared voters, the segment Mrs. Haley was hoping would turn out for her.

“I am voting for President Trump because I wholeheartedly believe that if he was still in office Title 42 would not have expired, and when he was president the value of the American dollar was never higher,” an undeclared voter in his late 20s, Matthew Paul, told the Sun.

“Trump never fit in, and he always said he wanted to drain the swamp so I’m supporting him for that,” a registered undeclared voter and lobsterman, Justin Robert, told the Sun.

“We need a bull in the china closet so to speak, and Trump is that,” a Republican and retired FBI agent, Ralph Gault, told the Sun. “He takes a strong position and this country, international issues, the dangers of our enemies, we just need somebody who will stand up to all this.”

On the Democratic side, President Biden won the primary handily, despite not being on the ballot. On Tuesday evening, results from the AP showed Mr. Biden with some 52 percent of the vote, while his most formidable challenger, the three-term congressman from Minnesota, Dean Phillips, trailed with 20 percent. Self-help author Marianne Williamson earned a little less than 5 percent.  Some 14 percent of the votes were unprocessed write-in ballots as of Tuesday evening.

Mr. Biden’s name was not on the ballot because New Hampshire refused to comply with Democratic National Committee primary calendar changes. The DNC declared New Hampshire’s primary “meaningless” and said it would not be used to award delegates. Mr. Biden did not campaign in the state.

Yet Biden supporters and campaign surrogates launched a robust “Write-in Biden” campaign, sending out mailers and hosting get-out-the-vote events, to make sure the incumbent did not suffer an embarrassing loss. Volunteers held “Write-in Biden” signs at the polling sites the Sun visited. The effort worked.

“This is a heavily Democratic ward, and so we’re getting lots of thumbs up from people here,” a Write-in Biden volunteer, Ellen Fineberg, told the Sun outside a Portsmouth polling site. “This is a very confusing election.”

Mr. Phillips, though, told CNN Tuesday morning that he expected to get in the 20-percent range and would not be dropping out of the race. “There is an immense need to keep this challenge going and I will continue it,” Mr. Phillips said. “Joe Biden should be at 80 percent, even with a write in.”

Mr. Phillips also called on Democrats with more name recognition, such as J.B. Pritzker, Gretchen Whitmer, and Gavin Newsom, to jump in the race. “Joe Biden is going to lose,” Mr. Phillips said. “My party is completely delusional right now, and somebody had to wake us up.”

At New Hampshire polling sites, the Sun spoke with several Phillips voters who expressed concern for Mr. Biden’s age and frustrations with Mr. Biden for blackballing the state. “I think people are really upset that Biden chose not to put his name on the ballot,” New Hampshire’s secretary of state, David Scanlon, told the Sun.

“I am a true believer in New Hampshire’s first-in-the-nation primary, and I think it was unfair for the national Democratic Party to ignore that,” a registered Democrat and restauranteur in Portsmouth, Jack Blalock, told the Sun. “So I voted for someone actually on the ballot.”

Mr. Blalock called his Phillips vote “a protest vote,” but he is hoping a younger Democrat will step up to challenge Mr. Biden.

“I don’t like Biden’s stance of ‘I’m not going to support you in New Hampshire but I want you to support me,’ so I went with Dean,” an retired executive, Neil Garvey, told the Sun. “If it’s Trump-Biden I don’t know what I’m going to do.”

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This story was updated to reflect the latest vote results provided by the AP.