Private Mediterranean islands rarely come up for sale, but Punta Pennata, a sliver of land in the Gulf of Naples with the crumbling walls of an old Roman villa tucked amid its overgrown flora, has been on the market for several months.

Located about 20 miles from downtown Naples, the narrow island with its abrupt cliffs sits like an exclamation point just off the seaside town of Bacoli.

“Surrounded by lush Mediterranean vegetation and Roman relics, it offers a fascinating and unique retreat,” reads the promotional blurb from Sotheby’s International Realty. Punta Pennata, it adds, is an “exclusive opportunity for a prestigious investment.”

That is one version of the island’s future.

The other comes from Josi Gerardo Della Ragione, the mayor of Bacoli, who envisions the island as a public park. In private hands for decades and barely visited, its unspoiled flora would draw residents and visitors alike to a lesser known part of Italy’s coastline, according to the mayor.

“It is like Neverland,” he said in an interview in his office, evoking Peter Pan’s fictional island. “It is a place you can go just to daydream,” he added. “Bacoli is small, but it can still be chaotic.”