Curtis Sliwa has six cats, wears a red beret and is running to be the next mayor of New York City. Many wish he wouldn’t.

President Trump has dismissed the Republican candidate as “not exactly ready for prime time.” Billionaire Bill Ackman has urged him to quit the race “for the good of NYC.” Sliwa, trailing in the polls, says he gets calls from people telling him it’s time to go. He isn’t budging.

“Curtis Sliwa never dropped out of anything in his life,” he said in front of a Manhattan subway station recently. “Any questions?”

A wild card in one of the wildest mayoral races in New York history, Sliwa says he is the one who can beat Democratic nominee and front-runner Zohran Mamdani —even though in his 2021 mayoral campaign the Republican only captured about 28% of the votes in an overwhelmingly blue city.

Critics fear his candidacy will siphon potential crossover votes away from Andrew Cuomo , who is campaigning as the anti-socialist option against Mamdani ahead of the Nov. 4 election. Mamdani, who decisively beat Cuomo in the Democratic primary, has been polling well ahead in the most recent surveys.

The possibility of a Mamdani win worries business leaders such as Ackman, who don’t want to see some of his campaign promises—including rent freezes and city-run supermarkets—come to fruition. The pressure on Sliwa increased when Mayor Eric Adams   dropped his re-election campaign last month. With the departure of Adams, both Cuomo and Mamdani—and their respective backers—eagerly embraced the idea of a two-man race.

But standing in their way is 71-year-old Sliwa, a Guardian Angel and subway vigilante who cemented his place in city lore after surviving five gunshot wounds in an alleged mob hit in 1992.

“People know Curtis,” said Ed Cox, chairman of the New York Republican Party. “In a parade, they see the red beret, they cheer. They know for 45 years he’s been working for them.”

Sliwa’s favorite campaign spot is the subway, which the former Bronx McDonald’s manager started patrolling in the 1970s after founding his anticrime organization Guardian Angels. When someone recognizes him, he shakes their hand and gives them a business card.

“Curtis is the only candidate for NYC mayor who rides the subway,” it says. “Just like you.”

Crime is the top issue for subway riders, according to Sliwa’s informal chats. If elected, Sliwa said he would hire 7,000 more police officers and do more to end building package theft and drugstore shoplifting.

“Why do we live in a city where we lock up toothpaste and we don’t lock up criminals?” he said. But New York City’s crime doesn’t warrant federal intervention, Sliwa said, and he doesn’t support having President Trump send in troops as he has done in places such as Memphis, Tenn.

He backs Trump’s crackdown on criminals who are in the U.S. without authorization, but doesn’t approve of mass deportations for people who haven’t committed any crimes.

“That’s not what the public is in favor of,” said Sliwa. “What are we going to do, close all the restaurants? Are we going to close the hospitality industry? Many of these are essential workers.”

He still wears the red beret from his Guardian Angels heyday because “it’s how people pick you out in a crowd.” But if elected, it won’t come with him to Gracie Mansion, the mayor’s official residence. “It’ll be put into storage,” he said.

Along with straphangers, Sliwa is targeting animal lovers for votes. He has promised to turn city animal shelters into no-kill havens for homeless pets, and has said he’ll ban carriage horse rides in Central Park—a campaign promise former Democratic Mayor Bill de Blasio made ahead of his first term but failed to achieve during eight years in office.

Sliwa leans on his wife, Nancy Sliwa, who is 30 years his junior, to overcome the age gap between him and 33-year-old front-runner Mamdani, whose popularity among young voters has been the main driver of his success in the race.

Over the summer, Sliwa went to an outdoor electronic dance music concert under a Brooklyn bridge and shared a video on X of himself in a sea of 20-somethings with his hand in the air and his red-bereted head bobbing. He posted it to show voters he isn’t “an old fuddy-duddy,” he said. “They can relate to you a little bit.”

During his neighborhood walkabouts, Sliwa willingly stops to pose for photos with young people, part of his strategy to build visibility on social media.

“Suddenly you’re on their Instagram, they’re commenting about you—nice stuff,” said the Brooklyn-born Sliwa, who has remained a New York fixture for decades through frequent TV appearances and as the host of a conservative talk radio show where he frequently railed about city politics.

Along with fighting crime, Sliwa’s campaign promises include a novel concept for tackling the city’s ballooning rat problem. A longtime cat lover—he lives with his wife and a half-dozen cats in a one-bedroom Manhattan apartment—Sliwa wants to weaponize the city’s feral felines.

“Consider them like Batman and Robin—Gotham caped crusaders at night,” he said at an August press conference detailing his plan to bring cat colonies into Central Park to let nature take its course against the rats. During Sliwa’s first mayoral campaign, he made headlines for keeping 17 cats in his then-320-square-foot studio apartment.

Of his current cat companions, Sliwa said they’re his family. One of them, Ajax, is neurologically impaired. “He’ll walk a few steps and then fall down,” he said. Mittens, who struggles with weight issues, “is hoping to get Ozempic.”

Sliwa’s campaign platform hasn’t exactly silenced the chorus of naysayers who doubt he can beat Mamdani by running on cats and crime. But Sliwa believes he has one more thing in his favor: a confusing general election ballot.

His name is on the ballot twice, first as a Republican, and then as an independent on the Protect Animals line. He appears ahead of Cuomo. The ballot also lists Adams and former independent candidate Jim Walden—they dropped out too late to be removed—and two lesser-known contenders for a grand total of seven names.

“It’s like you’re going through a maze,” Sliwa said of the ballot. “Not easy.”

One Democrat, 72-year-old Jacqueline Pollen, said she is up for the challenge. She still remembers Sliwa’s past efforts to combat subway crime and plans to vote for him.

“I felt very safe for many years because of that,” Pollen said, as she stopped to listen to Sliwa at a Manhattan campaign stop.

Moments later, another onlooker approached Sliwa and urged him to drop out. Sliwa shook his hand, handed him a card and gave his standard reply: No.

Write to Joseph Pisani at joseph.pisani@wsj.com