Add water and raspberries to a food processor. Pulse until mixture has a smooth consistency.
Place a fine mesh strainer over a large bowl. Add pureed raspberries to the strainer (1/4 at a time) and press with the back of a large spoon or a rubber spatula through the strainer to remove the seeds. Repeat with remaining raspberry puree. Discard anything left in the strainer.
Add sugar, vanilla, and lime juice to the raspberry juice in the bowl. Whisk until sugar is dissolved.
Freeze according to your ice cream maker's instructions. Serve immediately for a soft-serve dessert OR transfer to a shallow container and freeze an additional 1-2 hours for a more firm sorbet.
Store in airtight container in freezer up to 1 month.