Flying robots are usually rather delicate and require protective enclosures when facing the risk of collision, while high complexity and reduced payload are recurrent problems with collision-resilient flying robots. Inspired by arthropods' exoskeletons, we design a simple, open source, easily manufactured, semi-rigid structure with soft joints that can withstand high-velocity impacts. With an exoskeleton, the protective shell becomes part of the main robot structure, thereby minimizing its loss in payload capacity. Our design is simple to build and customize using cheap components (e.g. bamboo skewers) and consumer-grade 3D printers. The result is CogniFly, a sub-250g autonomous quadcopter that survives multiple collisions at speeds up to 7m/s. In addition to its collision-resiliency, CogniFly is easy to program using Python or Buzz, carries sensors that allow it to fly for approx. 17min without the need of GPS or an external motion capture system, has enough computing power to run deep neural network models on-board and was designed to facilitate integration with an automated battery swapping system. This structure becomes an ideal platform for high-risk activities (such as flying in a cluttered environment or reinforcement learning training) by dramatically reducing the risks of damaging its own hardware or the environment. Source code, 3D files, instructions and videos are available through the project's website (https://thecognifly.github.io).