Biological sensing and processing is asynchronous and sparse, leading to low-latency and energy-efficient perception and action. In robotics, neuromorphic hardware for event-based vision and spiking neural networks promises to exhibit similar characteristics. However, robotic implementations have been limited to basic tasks with low-dimensional sensory inputs and motor actions due to the restricted network size in current embedded neuromorphic processors and the difficulties of training spiking neural networks. Here, we present the first fully neuromorphic vision-to-control pipeline for controlling a freely flying drone. Specifically, we train a spiking neural network that accepts high-dimensional raw event-based camera data and outputs low-level control actions for performing autonomous vision-based flight. The vision part of the network, consisting of five layers and 28.8k neurons, maps incoming raw events to ego-motion estimates and is trained with self-supervised learning on real event data. The control part consists of a single decoding layer and is learned with an evolutionary algorithm in a drone simulator. Robotic experiments show a successful sim-to-real transfer of the fully learned neuromorphic pipeline. The drone can accurately follow different ego-motion setpoints, allowing for hovering, landing, and maneuvering sideways$\unicode{x2014}$even while yawing at the same time. The neuromorphic pipeline runs on board on Intel's Loihi neuromorphic processor with an execution frequency of 200 Hz, spending only 27 $\unicode{x00b5}$J per inference. These results illustrate the potential of neuromorphic sensing and processing for enabling smaller, more intelligent robots.