Preheat the oven to 350°F [180°C]. Butter and flour a 9-by-5-by-3-in [23-by-12-by-7.5-cm] loaf pan.
In a small bowl, combine the all-purpose flour, whole-wheat flour, baking soda, and salt. Set aside.
In a large bowl, with a hand mixer, beat the butter and sugar until light and fluffy, about 3 minutes. Beat in the bananas and the eggs, one at a time, until well combined but still chunky. Gradually add the flour mixture and mix just until combined. Fold in the walnuts and chocolate chips. Pour the batter into the prepared loaf pan.
Bake for 55 to 65 minutes, or until a toothpick inserted into the center comes out clean. Let the bread cool in the pan for 15 minutes. Loosen the bread from the pan by running a knife around the sides. Invert a wire rack over the top of the pan. Using both hands, flip the bread and rack over so the loaf pan is upside down on the rack. Gently shake the pan to release the bread onto the rack. Flip the bread back over so it is right-side up. Allow to cool completely, then wrap tightly in plastic wrap. Store wrapped tightly in plastic wrap at room temperature for up to 5 days; or in the freezer for up to 2 months. To defrost, move the wrapped bread to the refrigerator overnight. Reheat in a 350°F [180°C] oven for 5 to 10 minutes or until heated through.