Preheat oven to 325°F. Lightly spray a 8x5-inch loaf pan with nonstick spray and dust with flour.
In a large bowl, use an electric mixer to beat the eggs and sugar until thick and light, about 5 minutes. Mix in the bananas, buttermilk, oil and vanilla.
Sift the flour, baking powder, baking soda and salt over the wet mixture; beat just until blended. Transfer the batter to the prepared pan.
Bake the bread until golden brown on top and a tester inserted into the center comes out clean, about 1 hour. Turn the bread out onto a rack and cool.