Bring the water to a roaring boil in a medium size pot. Add tapioca pearls, stir occasionally.
To make the banana puree – with the skin on, slice the banana in half, lengthways.
With a small teaspoon, gently scrape down the middles of each side, removing the seeds, but leaving the flesh ( this will keep the banana from turning brown ) Discard the skin, and blend the flesh in a food processor until smooth.
Bring banana puree, milk, and sugar to a boil in a separate pot.
Once tapioca is almost cook, strain into a fine sieve. Rinse well with cold water to remove the starch and stop the pearls sticking together and becoming stodgy.
Add to pot with puree mixture and finish cooking. Pour into plastic container, cover, refrigerate and let cool overnight.
Add lime zest and salt to taste.
Serve in a small bowl or glass, like a martini or whiskey glass.