Henri Bendel, the specialty retailer known for its signature brown-and-white bags, is shutting down after more than a century in business.

L Brands, which acquired the retailer in 1985, said it would close all 23 Henri Bendel stores, including its Fifth Avenue location, and its website, after the holiday season. The company said it plans to focus on its larger businesses including Victoria’s Secret, which has struggled in recent quarters.

“We...