Detecting and estimating distances to power lines is a challenge for both human UAV pilots and autonomous systems, which increases the risk of unintended collisions. We present a mmWave radar-based perception system that provides spherical sensing coverage around a small UAV for robust power line detection and avoidance. The system integrates multiple compact solid-state mmWave radar modules to synthesize an omnidirectional field of view while remaining lightweight. We characterize the sensing behavior of this omnidirectional radar arrangement in power line environments and develop a robust detection-and-avoidance algorithm tailored to that behavior. Field experiments on real power lines demonstrate reliable detection at ranges up to 10 m, successful avoidance maneuvers at flight speeds upwards of 10 m/s, and detection of wires as thin as 1.2 mm in diameter. These results indicate the approach's suitability as an additional safety layer for both autonomous and manual UAV flight.