Dianna Carlin should be finishing the book she is writing about the joys of owning the Lola Star boutique, a “really tiny, magical little shop” on the Coney Island boardwalk, for the past 19 years.

Instead, Ms. Carlin has been anxious and fearful since her landlord weeks ago offered her a new lease — with a 400 percent rent increase. “I’m wondering if I should start ordering ‘going out of business’ signs,” she said.

In the summer, Brooklyn’s Coney Island swells with sunbathers and amusement seekers, but it tends to be much quieter in the winter.

Over a decade ago, New York City promised a year-round destination with a water park, an arena, an ice-skating rink, and millions of dollars in residential and commercial investment.