Preheat oven to 325°F. Lightly grease 8 1/2x4 1/2x2 1/2-inch pan; dust with flour. Using electric mixer; beat eggs and sugar in large bowl until thick and light, about 5 minutes. Mix in smashed bananas, buttermilk, oil and vanilla. Sift flour, baking powder, baking soda and salt over mixture; beat until just blended. Transfer batter to prepared pan.

Bake bread until golden brown on top and tester inserted into center comes out clean, about 1 hour. Turn bread out onto rack and cool.