Lightly grease a 6cm-deep, 10cm x 20cm (base) loaf pan. Line the base and sides with baking paper, allowing a 2cm overhang at long ends.
Sift flour and bicarbonate of soda into a bowl, add sugar and stir to combine. Place milk, eggs and vanilla in a jug and whisk to combine.
Fold milk mixture and banana into flour mixture and pour into a prepared pan.
Bake for 1 hour to 1 hour 15 minutes or until a skewer inserted in the centre of the cake comes out clean.
Stand cake in pan for 10 minutes.
Turn out on to a wire rack to cool.
To make the icing: Place the brown sugar, butter and milk in a saucepan over low heat. Cook, stirring, for 2 minutes (don't boil) or until sugar dissolves. Increase the heat to medium. and bring to boil.
Reduce heat to low. and cook for 2 to 3 minutes or until slightly thickened.
Remove from heat. Stir in icing sugar until smooth.