Pocket-sized autonomous nano-drones can revolutionize many robotic use cases, such as visual inspection in narrow, constrained spaces, and ensure safer human-robot interaction due to their tiny form factor and weight -- i.e., tens of grams. This compelling vision is challenged by the high level of intelligence needed aboard, which clashes against the limited computational and storage resources available on PULP (parallel-ultra-low-power) MCU class navigation and mission controllers that can be hosted aboard. This work moves from PULP-Dronet, a State-of-the-Art convolutional neural network for autonomous navigation on nano-drones. We introduce Tiny-PULP-Dronet: a novel methodology to squeeze by more than one order of magnitude model size (50x fewer parameters), and number of operations (27x less multiply-and-accumulate) required to run inference with similar flight performance as PULP-Dronet. This massive reduction paves the way towards affordable multi-tasking on nano-drones, a fundamental requirement for achieving high-level intelligence.