Combine 1 cup milk, bananas, and vanilla extract in a blender or food processor. Blend until very smooth and creamy, about 60 seconds on a blender's full speed. If the blades are catching, add more milk, just a splash at a time, to get the ingredients moving. Serve immediately, or transfer to a shallow dish and freeze for 30 minutes to firm it up and make it more scoopable (in a deep container it will take longer).
Transfer to an airtight container to freeze any leftovers, then let stand at room temperature for 15 minutes or so to soften before breaking up ice crystals with a fork, stirring well, and then scooping.
To make strawberry sauce, give the blender a good rinse, then add the water and strawberries to the container. Blend on high speed until very smooth. Add more water if needed to keep the ingredients moving. Refrigerate any leftovers in an airtight container.