Now, Mr. Heniche, who is based in Milan, Italy, says demand is so charged he regularly faces bidding wars. Some prospective sellers have so many offers they have taken to organizing them in Excel spreadsheets. Those looking to sell lightly used Model 3s can expect to pocket upward of $4,500 in profit, he said.

“I’ve never seen anything like this before,” said Mr. Heniche, 26, who has been selling cars for about eight years. He got his nickname, Tesla Pimp, from one of his suppliers.

The car market has taken a sharp turn: Some used cars, including lightly used Teslas, cost more than new ones. People can’t get certain new models because of parts shortages and other supply-chain bottlenecks. That’s driving up prices of used vehicles—and upending the adage that cars lose value the minute you drive off the lot.

Wait times for new Teslas in the U.S. range from weeks for the Model S Plaid sedan to roughly a year for one configuration of the Model X sport-utility vehicle.

One-year-old Model Ys, one of Tesla’s most-popular models, sold for an average of more than $65,000 in December, some $2,100 more than commonly configured new versions of the compact sport-utility vehicle, according to U.S. sales data provided by J.D. Power.

Tesla, which didn’t respond to a request for comment, is an extreme example of the turbocharged market for used vehicles.

“It’s shocking,”

Jim Farley,

chief executive of Ford Motor Co., said of the used-car market in a recent interview.

Lightly used versions of other models, such as the Toyota RAV4 and Kia Telluride, are also regularly selling above sticker prices, according to car-shopping website Edmunds.com.

The appetite for used Teslas has driven some owners to a fork in the road. Michael McKee paid more than $48,200 for his new Tesla Model 3 car which he picked up last March. Within seven months, a dealer was offering more than $50,000 to buy it from him.

Mr. McKee, a 42-year-old project manager, wasn’t looking to sell the car. But, with hopes of retiring early, he was on the lookout for opportunities to pad savings. A dealer eventually offered around $55,600 for the car, which had some 21,500 miles on it. “I just couldn’t pass it up,” he said.

Mr. McKee, who now drives a 2015 Hyundai Sonata, said he lost more than just a comfortable ride when he parted with his Tesla. The car was also his on-ramp to a tightknit group of Tesla fans. “If you ever saw someone driving a Tesla where I’m at, you’d wave. You’d park next to them in a parking lot,” he said. His current ride, he said, is “just a boring old car.”

When Amit Malhotra, a 48-year-old entrepreneur, went to get the bumper on his 2018 Model 3 fixed last fall, his mechanic suggested he see what the local Texas dealership would give him for the car.

“This is one of your quintessential he went out to get bread and sold my Tesla” stories, said his wife, Radhika Malhotra, a 48-year-old mother of two.

Ms. Malhotra said she remembers getting her husband’s call: “Hey, can I sell this car now? We’re getting a fantastic deal.” The answer to her was obvious. “No…You can’t just go out and literally sell our car,” she told him.

She demanded her husband right then order a replacement Model Y and deliver the proof before going through with the sale.

Their new Tesla is set to be delivered this summer.

Waiting has been tough for Ms. Malhotra, who loved how her Tesla’s driver-assistance system served almost as a backup driver. “In the beginning, honestly it was like someone coming off a sugar addiction,” she said.

Juan Miranda, a 39-year-old mechanical test engineer in San Diego, didn’t want to wait until July to secure his family’s second Model Y. So in December, he tried to shell out roughly $71,600 for a used one with nearly 10,000 miles—several thousand dollars more than he paid earlier in the year for a new Model Y with more bells and whistles.

But Tesla, which operates a market for used versions of its cars, would only hold the vehicle for three days, said Mr. Miranda. He had sold Tesla stock to buy the vehicle and couldn’t get the cash together that quickly. By the time he was ready, the car was gone and Tesla didn’t have others available that fit his criteria.

“A couple of times I found one, I clicked ‘buy now’ and then it was gone,” said Mr. Miranda, who spent weeks searching for another Model Y. “I’d call them and they’d be like ‘oh yeah that one got snatched up’ and I was like ‘dammit.’ ”

As the hunt dragged on, he grew exasperated. “This is ridiculous. Can someone just sell me a used Model Y?” he fumed, before finally settling for buying a used Tesla Model 3.

The used-car market is forecast to remain hot before softening toward the end of 2022, when supply-chain bottlenecks are expected to ease, said Ivan Drury, an analyst for Edmunds.com.

Teslas have long retained their value relatively well. That’s in part because the company invested in battery capacity and electronics early on, enabling it to upgrade older vehicles in a way traditional car makers couldn’t, said

Mark Wakefield,

a managing director at consulting firm AlixPartners LLP.

Shane Andrews waited 18 months or more for her family’s Teslas in the past and didn’t want to do that again. She signed on with a car-leasing company, Autonomy, that launched late last year in Southern California and she is leasing a Model 3. “We really needed something quickly,” said Dr. Andrews, who wanted another car so that she could give her daughter, then about to turn 16, one of the family’s older vehicles.

For Mr. McKee, who still regrets selling his Model 3 to pad savings, the love affair with Tesla isn’t over. One day, in retirement, he aims to again own a Tesla.

Write to Rebecca Elliott at rebecca.elliott@wsj.com and Meghan Bobrowsky at Meghan.Bobrowsky@wsj.com