Just 12 years after declaring its independence, Montenegro is gaining traction in its ambitious goal of becoming one of the dominant players in the Balkans luxury-resort business.

The tiny country’s first celebrity-endorsed golf course—designed by Gary Player—is scheduled to launch next month as part of Luštica Bay, Montenegro’s biggest resort development. That same day, Luštica Bay’s developer, Egyptian firm Orascom Development Holding, is planning to open the resort’s first five-star hotel, the Chedi Luštica Bay, with prices...