Flying robots that can locomote efficiently in GPS-denied cluttered environments have many applications, such as in search and rescue scenarios. However, dealing with the high amount of obstacles inherent to such environments is a major challenge for flying vehicles. Conventional flying platforms cannot afford to collide with obstacles, as the disturbance from the impact may provoke a crash to the ground, especially when friction forces generate torques affecting the attitude of the platform. We propose a concept of resilient flying robots capable of colliding into obstacles without compromising their flight stability. Such platforms present great advantages over existing robots as they are capable of robust flight in cluttered environments without the need for complex sense and avoid strategies or 3D mapping of the environment. We propose a design comprising an inner frame equipped with conventional propulsion and stabilization systems enclosed in a protective cage that can rotate passively thanks to a 3-axis gimbal system, which reduces the impact of friction forces on the attitude of the inner frame. After addressing important design considerations thanks to a collision model and validation experiments, we present a proof-of-concept platform, named GimBall, capable of flying in various cluttered environments. Field experiments demonstrate the robot's ability to fly fully autonomously through a forest while experiencing multiple collisions.