In a blender, combine 1 1/2 cups blackberries, the sugar, and lime juice. Blend until smooth. Pour through a fine-mesh sieve into a pitcher, rubbing to extract juices, then stir in gin.
Half-fill 6 tall glasses with ice cubes. Divide the gin mixture evenly among the glasses. Fill each with sparkling water. Garnish each glass with a few of the remaining blackberries and a slice or two of lime.