In a blender, combine the blackberries and water. Blend until pureed. Strain the juice through a fine-mesh sieve and discard all the seeds.
In a glass, add the mint leaves, juice from half a lime and the lime rind, and a splash of the soda water. Gently muddle to release the flavour from the mint leaves. Fill the glass with ice and add the rum, blackberry juice, and top with soda water. Stir to combine. Garnish with lime wedge, mint sprig and blackberries.