Panama’s Pearl Islands have long held hidden treasures. The archipelago of some 39 islands and 100 islets, located about 56 miles off the coast of Panama City, has yielded some of the world’s most precious pearls, including La Pelegrina, a 111-grain silver pearl coveted by 19th-century Russian royalty, and La Peregrina, a pear-shaped specimen that made its way through a long line of kings and queens before settling around the neck of Elizabeth Taylor. The region’s latest gem, however, is an island itself—a 3,7…