Line a large baking sheet with parchment or wax paper. Arrange diced papaya in a single layer and freeze overnight.

Combine water and sugar in a small saucepan. Bring to a boil over high heat, stirring constantly, until the sugar is completely dissolved and syrupy, 3 to 5 minutes. Pour the syrup into a glass measuring cup and let cool for 15 minutes. Remove the papaya from the freezer to defrost while the syrup is cooling.

Transfer half the frozen papaya to a food processor. Add half the syrup, 2 tablespoons coconut milk, 1 tablespoon lime juice, 1/2 teaspoon lime zest and a pinch of salt. Pulse two or three times then process until smooth, stopping to scrape down the sides and stir as necessary. Transfer the sorbet to a large serving bowl. Repeat with the remaining ingredients; add to the serving bowl. Serve immediately or freeze until ready to use.

Ingredient Note: Very large papayas, 3 to 4 pounds each, are available in many big supermarkets. Smaller papayas (about the size of a mango and yielding about 1 cup of diced fruit each) will also work well. To select a ripe papaya, look for skin that is about half golden yellow and half green and yields to gentle pressure, like a ripe peach or nectarine. It's common to find very green papayas in the store, but once home, they'll ripen - out of the refrigerator - in 2 to 3 days.

MAKE AHEAD TIP: The sorbet will keep in the freezer, without freezing solid, for up to 3 hours. (Alternatively, prepare through Step 3, transfer to a shallow pan, cover and freeze for up to 1 month. Before serving, let defrost for 30 minutes, break into small chunks and process in small batches in a food processor until smooth and scoopable.)