Put the litchi in a pan with the water and sugar, and bring to a boil. Mix well, making sure the sugar has dissolved fully. Then allow it to cool.
When the compote has cooled, deseed the litchis and keep aside the syrup.
Mix the litchi flesh with the sugar syrup and blend in the lime juice. Chuck them into the food processor and blitz until everything is chopped and of even consistency. Then if you like, press everything through a sieve for extra smoothness.
Freeze the mixture until solid.
Once solid, smash up the sorbet and churn again for 5-10 minutes. Freeze overnight and then serve.