Nano-size drones hold enormous potential to explore unknown and complex environments. Their small size makes them agile and safe for operation close to humans and allows them to navigate through narrow spaces. However, their tiny size and payload restrict the possibilities for on-board computation and sensing, making fully autonomous flight extremely challenging. The first step towards full autonomy is reliable obstacle avoidance, which has proven to be technically challenging by itself in a generic indoor environment. Current approaches utilize vision-based or 1-dimensional sensors to support nano-drone perception algorithms. This work presents a lightweight obstacle avoidance system based on a novel millimeter form factor 64 pixels multi-zone Time-of-Flight (ToF) sensor and a generalized model-free control policy. Reported in-field tests are based on the Crazyflie 2.1, extended by a custom multi-zone ToF deck, featuring a total flight mass of 35g. The algorithm only uses 0.3% of the on-board processing power (210uS execution time) with a frame rate of 15fps, providing an excellent foundation for many future applications. Less than 10% of the total drone power is needed to operate the proposed perception system, including both lifting and operating the sensor. The presented autonomous nano-size drone reaches 100% reliability at 0.5m/s in a generic and previously unexplored indoor environment. The proposed system is released open-source with an extensive dataset including ToF and gray-scale camera data, coupled with UAV position ground truth from motion capture.