Preheat oven to 350°F. Grease a 9x5-inch loaf pan with nonstick spray or butter, then line it with parchment paper, leaving an overhang on the long sides.
Peel 4 medium bananas and place them in a bowl. Mash with a fork until mostly smooth, leaving a few small lumps, and set aside.
In a large bowl, cream together 2/3 cup unsalted butter, softened, and 3/4 cup granulated sugar with a hand mixer on medium speed (or use a wooden spoon) until light, pale, and fluffy.
Beat in 2 large eggs one at a time, mixing well after each addition. If the batter looks curdled, stir in 1 tablespoon of flour and continue beating.
In a separate bowl, sift together 1 7/8 cups self-rising flour, 1 teaspoon baking soda, and a pinch of salt.
Add the sifted dry ingredients, mashed bananas, and 1 teaspoon vanilla extract to the butter mixture. Gently fold with a spatula until no streaks of flour remain.
Transfer the batter to the prepared loaf pan and smooth the top. Bake at 350°F for 50–60 minutes, or until the bread is golden and a toothpick inserted into the center comes out clean. If the top browns too quickly, loosely tent with aluminum foil for the last 10–15 minutes.
Let the loaf cool in the pan for 10–15 minutes. Then, using the parchment overhang, lift it out and transfer to a wire rack to cool completely.