Mash the bananas in a medium bowl. Add the peanut butter and applesauce to the mashed bananas and stir until well combined. Set this wet mixture aside.
In a separate bowl, whisk together the gluten-free flour and baking soda.
Gradually add the wet ingredients to the dry ingredients, stirring constantly until just combined. Be careful not to overmix.
Line a loaf pan with parchment paper and pour the batter into the prepared pan.
Bake at 350°F (175°C) for 50 minutes, or until a toothpick inserted into the center comes out clean.
Let the bread cool in the pan for 10 minutes before transferring it to a wire rack to cool completely.