An illustrative example embodiment of an elevator governor includes a rotatable governor mechanism, a tension sheave, and a tension frame associated with the tension sheave. The tension frame has a mass configured to bias the tension frame and the tension sheave under an influence of gravity. A damper is configured to resist vertical movement of the tension frame relative to a fixed surface in a first condition and to allow vertical movement of the tension frame relative to the fixed surface in a second, different condition.