Preheat an oven to 350 degrees F (175 degrees C). Grease and flour a 9x5-inch loaf pan. Whisk the flour, sugar, baking soda, and salt together in a bowl.
Beat the egg in a mixing bowl. Whisk in the mayonnaise, bananas, and vanilla extract until evenly mixed. Stir in the flour mixture until no dry lumps remain. Pour the batter into the prepared loaf pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 45 minutes. Cool in the pan for 10 minutes before removing to cool completely on a wire rack.