Combine pudding mix and milk in a small bowl; stir with a whisk until smooth. Fold whipped topping into pudding mixture. Fold in bananas. Layer ¾ cup cookies in bottom of an 8-inch square baking dish. Spread half of pudding mixture over cookies. Repeat procedure with remaining cookies and pudding mixture. If desired, arrange sliced strawberries on top. Serve immediately, or cover and chill.