Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9x13 inch pan.
In a medium bowl, combine flour, baking soda and salt. In a large bowl, cream together butter and sugar until light and fluffy. Beat in eggs one at a time, then stir in vanilla. Beat in the flour mixture alternately with the mashed bananas.
Pour batter into prepared pan. Bake in preheated oven for 40 to 50 minutes, or until a toothpick inserted into the center of the cake comes out clean. Let cool in pan for 10 minutes, then turn out onto a wire rack and cool completely.