The city that never sleeps has good reason to remain sleepless these days. A new terror imperils New York, threatening to destroy all that it — nay, America — holds dear.

No, it’s not a bomb, underground al-Qaida network, hurricane, alien invasion, asteroid, animatronic Stay Puft Marshmallow Man or any other favorite bugaboo of the silver screen. Nor is it even the latest stock market panic.

It’s boobs.

Yes, boobs. Bare breasts bouncing in the breeze in Times Square, to be specific. New York City has recently lost its mind because a few women have lost their shirts.

In Times Square, a handful of topless women have joined the hordes of buskers, proselytizers, balloon artists and costumed Disney characters hustling for tips. Wearing nothing but thongs, feather headdresses and star-spangled body-paint across their otherwise exposed chests, these women pose for photos with gawking tourists and accept tips given in appreciation for being what they call “entertainers” or “ambassadors” of this great city.

“We’re just out here having fun, welcoming people to the city. Sometimes people ask us for bar and restaurant recommendations,” native New Yorker and self-described “painted lady,” Angel Bunting, 32, told me Sunday during a smoke break. (Some of the other women instead call themselves “desnudas,” from the Spanish word for “naked.”)

Unorthodox though their wardrobe choices may be, these women are on the right side of the law. More than two decades ago, the state’s highest court ruled that prohibiting women, but not men, from baring their chests in public amounted to discrimination on the basis of sex. Panhandling is likewise legal, and any attempt to shut the desnudas down would likely run into First Amendment problems.

None of this has stopped a major freakout from the New York Daily News, followed by infuriated comments from Police Commissioner William Bratton. He promised the city was investigating every possible legal avenue to expel the exposed women from Times Square.

In a radio interview, Bratton proposed tearing up the Times Square pedestrian plazas, even though they have been responsible for a 35 percent reduction in pedestrian injuries. Apparently a few innocent maimings are worth it, if they enable the city to blot out this inexcusable blight.

Mayor Bill de Blasio is seriously considering the idea and even commissioned a task force to determine available options. New York Gov. Andrew Cuomo has also expressed his displeasure with Times Square’s painted ladies and has pledged his support in cracking down on such “illegal” activities.

This is all, in a word, bizarre.

There is no shortage of problems afflicting New York, starting with its crumbling subway system, failing schools and strained police-community relations. Scratch that — there are a lot of problems in just Times Square alone. But somehow the Times Square emergency wasn’t sparked by its anti-Semitic Elmo; Woody accused of groping ; allegedly handsy Cookie Monster; that other Cookie Monster accused of shoving a toddler; alleged pot-smoking, molesting Super Mario; alleged police-walloping Spider-Man; pugilistic Hello Kitty and Minnie Mouse; or even the area’s biggest taste-offender, the perplexingly popular M&M Store.

No, what sparked a crisis in the Crossroads of the World was everyday, run-of-the-mill human anatomy.

Who knew New York was so puritanical? I mean, they’re just breasts. Half of adults — more than half, really, if you look around — have them. Why is their visibility now a public safety threat?

When I asked Bunting this question, she argued that she was just being used as a distraction from “real” problems de Blasio doesn’t want to deal with.

Her fellow painted lady, Saira Nicole, 29, suggested that it was all about patriarchal control: “The female body is a commercial entity. But they don’t want us commercializing our bodies ourselves; only the Gap or Victoria’s Secret gets to do that.”

But while it’s easy to blame men, a recent survey from YouGov found that women are actually more likely to say they’re offended by publicly exposed breasts, in every scenario the poll asked about: when a woman is nursing an infant in public, sunbathing in a park, walking on the sidewalk or appearing on a magazine cover, newspaper front page or late-night TV. In every scenario, too, women were more likely than men to say that exposed breasts should be illegal.

Women are taught — it’s not innate, as clearly evidenced by the many unfazed young children who ambled by the desnudas on Sunday — to hate women’s bodies. Maybe we can use this manufactured crisis to teach the next generation a different lesson.