Preheat your oven to 375°F. Line a muffin tin with silicone or muffin liners or grease the muffin cups.
In a medium bowl, combine the rolled oats and milk. Let them soak for about 10-15 minutes until the oats have absorbed some of the liquid.
In a large bowl, whisk together the flour, baking powder, baking soda, and salt.
In another bowl, whisk together the mashed bananas, oil or melted butter, egg, and vanilla extract.
Pour the wet ingredients into the bowl with the dry ingredients. Add the soaked oats as well. Stir until just combined; do not overmix.
Gently fold in the blackberries until evenly distributed throughout the batter.
Spoon the batter into the prepared muffin cups, filling each about two-thirds full.
Bake in the preheated oven for 18-22 minutes or until a toothpick inserted into the center of a muffin comes out clean.
Allow the muffins to cool in the muffin tin for a few minutes before transferring them to a wire rack to cool completely.