Place frozen berries and sweetener, if using, in a food processor or blender; process or pulse just until the fruit is roughly chopped.
Pour in the heavy cream, and continue to process just until everything is incorporated and smooth. Add more heavy cream if the mixture is too thick. Scrape down the sides with a spatula as needed.
Serve immediately or freeze for later.
Notes
If you'd like to use coconut cream, blend 1 can (14 ounces) of coconut cream with 3 cups of frozen berries.
Before placing it in the freezer, cover the ice cream with plastic wrap to keep ice crystals from forming on the top.