Place prepared mango in a blender or food processor. Add the lime juice, water, and sugar. Blitz well.
Taste-test your puree - it should taste like a thick and very delicious mango juice (note that if you're using a semi-ripe mango OR frozen mango, you may need to add more sugar). If you want your martinis to have a very smooth texture, strain the puree through a sieve.
Measure all the martini ingredients into a shaker and shake it up, or put ingredients into a blender and blitz a few seconds.
Pour out into a martini glass and garnish with a slice of lime or a pretty starfruit slice.