Pour the vodka into a spotlessly clean bottle. Peel the lime peel with a thin peeler. Add to the vodka with the mint leaves. Put the bottle in the fridge for 3 weeks to infuse.
Bring the sugar and 300 ml of water to the boil in a saucepan. Stir until the sugar crystals have dissolved. Allow the syrup to cool completely. Add to the vodka. Leave in the fridge for another week.
Peel the lychees and remove the stone. Purée the lychees with the hand blender until smooth. Pour the vodka through a sieve. Add the lychee purée and serve in cocktail glasses.
