Divide the blackberries and mint leaves evenly among four glasses.
Add the lime juice and sugar to each glass. Use a muddler or the back of a spoon to press the berries and mint until the juices release.
Pour 2 ounces of white rum into each glass and stir to combine the flavors.
Fill the glasses with ice cubes. Top each one with about 4 ounces of club soda.
Drop in a few extra blackberries and a mint sprig for garnish before serving.