Stir flour, sugar, and salt together in a large bowl. Cut in butter using a pastry cutter or 2 knives until the texture resembles coarse cornmeal, with butter pieces the size of small peas. Mix in water with a fork just until the dough comes together.

Toss berries, sugar, flour, and salt together in a bowl to make the filling. Set aside at room temperature.

Lightly butter or grease 14 muffin cups. Place some dough in each muffin cup and press it against the bottoms and sides to create tart shells that are 1/8- to 1/4-inch thick. Divide berry filling evenly among the tart shells.

Position a rack in the lower third of the oven and preheat to 425 degrees F (220 degrees C). Refrigerate tarts for 15 minutes while the oven preheats.

Bake tarts in the preheated oven until crust is golden brown and filling is bubbling, 15 to 18 minutes.