Preheat the oven to 350°F (175°C). Grease a 9×5 inch loaf pan or line it with parchment paper.
In a large mixing bowl, combine the mashed bananas, peanut butter, melted butter, and granulated sugar. Stir until smooth and well blended.
Add the egg and vanilla extract to the banana mixture and mix until fully incorporated.
In a separate bowl, whisk together the baking soda, salt, and all-purpose flour.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Pour the batter into the prepared loaf pan and smooth the top with a spatula.
Bake in the preheated oven for 55 to 60 minutes, or until a toothpick inserted into the center comes out clean.
Remove the bread from the oven and let it cool in the pan for 10 minutes.
Transfer the bread to a wire rack to cool completely before slicing and serving.