Preheat the oven to 180C/350F/Gas 4.
Sift the flour, bicarbonate of soda and salt into a large mixing bowl.
In a separate bowl, cream the butter and sugar together until light and fluffy.
Add the eggs, mashed bananas, buttermilk and vanilla extract to the butter and sugar mixture and mix well. Fold in the flour mixture.
Grease a 20cm x 12.5cm/8in x 5in loaf tin and pour the cake mixture into the tin.
Transfer to the oven and bake for about an hour, or until well-risen and golden-brown.
Remove from the oven and cool in the tin for a few minutes, then turn out onto a wire rack to cool completely before serving.