Preheat the oven to 350°F (175°C). Grease a 9×5-inch loaf pan or line it with parchment paper.
In a medium bowl, whisk together the all-purpose flour, baking soda, and salt. Set aside.
In a large bowl, use an electric mixer to beat the softened butter, peanut butter, and granulated sugar until creamy and smooth.
Add the eggs one at a time to the butter mixture, beating well after each addition.
Mix in the mashed bananas and vanilla extract until combined.
Gradually add the dry ingredients to the wet ingredients, alternating with the milk, beginning and ending with the dry ingredients. Mix until just combined; do not overmix.
Pour the batter into the prepared loaf pan and spread it evenly.
Bake for 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.