Start by making the syrup – this should be done well ahead of time so it has time to cool.
Put the raspberries in a pan with the sugar and water (leave a couple aside with some mint leaves to decorate). Squeeze the lime juice in on top. Muddle the mint leaves in your hands and then add them to the pan. Put on a medium heat. Use a whisk to break down the raspberries and gently whisk it all together. Heat to a simmer and let it reduce by about a third. Strain into a jar and allow to cool.
When you are ready to serve, fill your pitcher up with ice. Pour about 100ml of your raspberry syrup in and then top up with soda water (about a 1:5 syrup:soda ratio). Stir to properly mix it, decorate with the remaining raspberries and mint; serve immediately.