Preheat oven to 400 degrees F (200 degrees C). Grease cookie sheets. Sift the flour, baking powder, salt, and baking soda together, and set aside.
Cream the butter with the sugar until light and fluffy. Beat in the eggs and vanilla. Mix in the mashed bananas. Add the flour mixture, and stir until just combined. Stir in the chocolate chips. Drop by spoonfuls onto prepared cookie sheets.