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Preheat the oven to 350°F. Line a muffin tin with cupcake liners.
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Combine and sift into a bowl the rice flour, potato starch, tapioca starch, baking powder, baking soda, cinnamon, salt, and xanthan gum. In a separate bowl, combine the bananas, rice milk, and vanilla.
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In the bowl of a stand mixer fitted with the paddle attachment or in a bowl with a whisk, mix the eggs, maple syrup, brown sugar, and oil until well combined. Add the dry ingredients and mix for 30 seconds. Add the banana mixture and combine well. Gently fold in the berries.
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Pour the batter into the prepared tin, filling each liner a little more than three-fourths of the way, and bake until a toothpick comes out clean, about 22 minutes. Let the muffins rest in the tin for 5 minutes. Transfer muffins to a wire rack to cool completely.