Add blackberries, sugar, lime juice, mint leaves, and 1/2 cup water in a saucepan over high heat. Use a wooden spoon to smash the blackberries, stirring to combine.
Once boiling, reduce heat to medium and cook for about 5 minutes. Remove from heat and pour into a blender. Blend until well combined. Strain over a pitcher to remove seeds and solids.
Stir remaining water into the pitcher and fill with ice. Stir in vodka and serve immediately. Garnish glasses with fresh blackberries and mint leaves.
notes
To make this a mocktail, simply replace the vodka with water!