Autonomous navigation in obstacle-dense indoor environments is very challenging for flying robots due to the high risk of collisions, which may lead to mechanical damage of the platform and eventual failure of the mission. While conventional approaches in autonomous navigation favor obstacle avoidance strategies, recent work showed that collision-robust flying robots could hit obstacles without breaking and even self-recover after a crash to the ground. This approach is particularly interesting for autonomous navigation in complex environments where collisions are unavoidable, or for reducing the sensing and control complexity involved in obstacle avoidance. This paper aims at showing that collision-robust platforms can go a step further and exploit contacts with the environment to achieve useful navigation tasks based on the sense of touch. This approach is typically useful when weight restrictions prevent the use of heavier sensors, or as a low-level detection mechanism supplementing other sensing modalities. In this paper, a solution based on force and inertial sensors used to detect obstacles all around the robot is presented. Eight miniature force sensors, weighting 0.9g each, are integrated in the structure of a collision-robust flying platform without affecting its robustness. A proof-of-concept experiment demonstrates the use of contact sensing for exploring autonomously a room in 3D, showing significant advantages compared to a previous strategy. To our knowledge this is the first fully autonomous flying robot using touch sensors as only exteroceptive sensors.