In a shaker or directly in the glass, muddle the blackberries, mint leaves, lime juice, and sugar together until the berries release their juice and the mint smells amazing.
Add the white rum and give everything a good stir (or shake, if using a shaker).
Strain the mixture into glasses if you want a smoother sip, leave the seeds in if you like it rustic.
Fill glasses with crushed ice, then top each with sparkling water.
Garnish with extra blackberries, a mint sprig, and a lime slice. Serve chilled and enjoy immediately!