Combine 2 cups blackberries, sugar, and water in a small saucepan. Heat over medium until the berries soften and release their juices, about 5 minutes.
Remove from heat and strain the mixture through a fine mesh sieve into a bowl, pressing gently on the solids to extract the liquid. Discard the seeds and pulp.
Stir the lime juice into the strained blackberry syrup until fully blended.
Fill four glasses with ice cubes. Divide the blackberry mixture evenly among the glasses.
Top each glass with 1 cup sparkling water and stir gently to combine.
Add a few fresh blackberries and one mint sprig to each glass before serving.