Heat the blackberries, sugar, and water in a small pot over medium-high heat. While warming, smash with a potato masher, until a thick even consistency is reached, about 8 minutes.
Pour the blackberries through a sieve, into a bowl, pushing some flesh through with the juices, making sure to catch the seeds. Cool in the refrigerator for at least 10 minutes.
Fill a cocktail shaker with 2 tablespoons of blackberry puree, 2 tablespoons of the mint leaves, 2 ounces of gin, 1 tablespoon lime juice, and a handful of ice. Shake vigorously 5 to 6 times until the drink is ice cold. Strain the mixture into a tall glass. Top with 6 ounces of ginger ale and stir lightly. Repeat for the remaining 3 servings. Finish and serve with a scoop of sorbet or ice cream.