An electronic device may include a voltage standing wave ratio (VSWR) sensor disposed along a radio-frequency transmission line between a signal generator and an antenna. The VSWR sensor may gather VSWR measurements from radio-frequency signals transmitted by the signal generator over the transmission line. Control circuitry may identify a variation in the VSWR measurements over time and may compare the variation to a threshold value to determine whether an external object in the vicinity of the antenna is animate or inanimate. The control circuitry may reduce the maximum transmit power level of the antenna when the external object is animate and may maintain or increase the maximum transmit power level when the external object is inanimate. This may serve to maximize the wireless performance of the electronic device while also ensuring that the device complies with regulatory limits on radio-frequency energy exposure.