Put the raspberries, buttermilk and sugar syrup in a blender. Peel the banana, cut into slices, also put in the blender and puree everything finely. Pour the mixture into a bowl, cover and let cool well in the refrigerator (preferably overnight).
The next day, pour into an ice cream maker and stir into the ice cream according to the manufacturer`s instructions. Serve the ice cream with fresh raspberries and a little lemon zest if you like.
Note: for the sugar syrup, bring equal parts of water and sugar to the boil and stir until the sugar is dissolved (sugar syrup). You can make a large amount of it, and it will last for months in bottles.