The ability of robots to model their own dynamics is key to autonomous planning and learning, as well as for autonomous damage detection and recovery. Traditionally dynamic models are pre-programmed, or learned from external observations and IMU data. Here, we demonstrate for the first time how a task-agnostic dynamic self-model can be learned using only a single first-person-view camera in a self-supervised manner, without any prior knowledge of robot morphology, kinematics, or task. We trained an egocentric visual self-model using random motor babbling on a 12-DoF robot. We then show how the robot can leverage its visual self-model to achieve various locomotion tasks, such as moving forward, backward and turning, all without any additional physical training. The accuracy of the egocentric model exceeds that of a model trained using an IMU. We also show how a robot can automatically detect and recover from damage. We suggest that self-supervised egocentric visual self-modeling could allow complex systems to continuously model themselves without additional sensors and prior knowledge.