Put the lime, palm sugar and basil in your cocktail shaker. Muddle the limes with the end of a clean rolling pin, crushing the sugar and basil and getting the juice out of the limes.
Add the sugar syrup, lychee juice and gin.
Fill the cocktail shaker with ice and shake vigorously until a frost forms on the shaker. Have a taste and adjust to taste.
Either strain into chilled martini glasses or serve in rocks glasses over plenty of crushed ice.