Flying robots are becoming increasingly widespread, not only as remote‐vision agents but also as tools capable of physically interacting with our world. As robots fly closer to obstacles, the likelihood of collision rises. While collision resilience solutions have been proposed for multirotor technology successfully, drones which rely on wings for much higher efficiency cannot sustain impacts without damage. Based on biological cues from woodpeckers, this article proposes a design approach to impact management which revolves around two new components. Head‐on collisions are handled by the robot's fuselage, replicating the brain‐protecting function of a woodpecker as it pecks a tree. Meanwhile, wing‐strike damage is mitigated with shoulder‐like structures. It is shown how these components can be implemented using tensegrity structures—the most optimal strength‐to‐weight architecture element—permitting tunable stiffness of joints and collision resilience. Optimization and manufacturing of these challenging structures as fuselage and wing joints are discussed. The approach is validated through integration into a fully flight‐capable robot, which is impact‐tested in controlled conditions and in outdoor environments. This method shows how tensegrity structures can be employed in flying robotics and paves the way toward winged drones safely operating in cluttered, contact‐prone environments.