Cut the watermelon into cubes, discarding the rind and any white parts. Add the watermelon cubes and lime juice to a blender and blend until smooth, using a tamper if necessary. If your blender is small, work in batches to blend all the watermelon.

Pour the blended mixture through a fine-mesh strainer to remove any pulp, then transfer the strained juice to a jug.

Fill glasses or a large pitcher with crushed ice. Pour the watermelon lime juice into each glass or pitcher until it’s halfway full. Top with soda water to fill the remaining space. Garnish each glass with a sprig of fresh mint and a lime slice, then serve immediately.