Heat raspberries, sugar, and water in a saucepan over medium heat until the raspberries begin to release their juices, about five minutes. Mash the berries with a potato masher until the berries break down, then stir in the lime zest and juice.
Simmer until the mixture is slightly thickened, about five minutes.
Pour the raspberry mixture through a fine-mesh strainer set over a bowl, pressing on solids to extract the liquid. This works best done in batches so the seeds don’t clog up the bottom of the mesh strainer. You should end up with about 2 1/2 cups syrup. Discard the berry solids. Refrigerate the syrup until completely cool, at least 30 minutes or up to 1 week.
To serve, combine 1 cup club soda and 1 tablespoons syrup in tall, ice-filled glasses.