Combine berries, sugar, zest and lime juice in a small saucepan.
Heat the mixture over medium heat, stirring occasionally.
When the berries release their juices, remove from the heat. Puree using a hand blender or stand blender.***
Return the puree to the saucepan and whisk in the sour cream, cream and vanilla until combined.
Transfer to a container and chill in the refrigerator for at least one hour.****
Once chilled, freeze according to your ice cream maker's instructions to a soft-serve consistency. (It's a small batch so mine took about 15 minutes.)
Transfer to a freezer-safe container (these work great) and freeze for at least four hours.
Serve (top with more berries if you like) and enjoy!