Peel the papaya and slice in half lengthwise. Scoop out the seeds and fibrous part with a small spoon.
Give the fruit a rough chop and puree in a food processor or blender. Scrape down the sides of the bowl if necessary. You should have about a cup of puree.
Strain the puree through a mesh strainer, stirring to push the liquid through. I added water to mine to help it get through. Most of it should go through, you will be left with just a little fiber.
Add enough water to the puree to make 3 cups liquid. You can fiddle with this proportion and add more water if you like, but remember you'll be serving it over ice, and also may want to add some sparkling water later, so don't dilute it too much to start with. Stir in the juice of one lime. Cover and refrigerate until cold.
To serve, pour the cold papaya water over ice, and garnish with a slice of lime and a few berries if you like. You can top off your glass with seltzer water if you like bubbles.