Preheat oven to 450 degrees F (230 degrees C). Line a muffin tin with paper liners.
Mix flour, sugar, baking powder, and salt together in a large bowl.
Whisk sour cream, butter, eggs, milk, and vanilla extract together in another bowl. Stir into the flour mixture until blended. Fold blackberries into the batter.
Spoon batter into the muffin tin, filling each liner 2/3 full.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 18 to 20 minutes.
