The paper proposes a nonlinear control method for performing a backflip maneuver with a nano quadcopter. To perform the maneuver, first a feasible reference trajectory is designed that describes the intended state evolution. Then, the designed trajectory is precisely tracked by a nonlinear geometric controller that is able to track even highly challenging reference trajectories. The performance of the proposed method is evaluated and compared to a simple adaptive feedforward control strategy based on simulations and real-world experiments using Bitcraze Crazyflie nano quadcopters.