We study the notion of superfluid critical velocity in one spatial dimension. It is shown that for heavy impurities with mass \(M\) exceeding a critical mass \(M_c\), the dispersion develops periodic metastable branches resulting in dramatic changes of dynamics in the presence of an external driving force. In contrast to smooth Bloch Oscillations for \(M<M_c\), a heavy impurity climbs metastable branches until it reaches a branch termination point or undergoes a random tunneling event, both leading to an abrupt change in velocity and an energy loss. This is predicted to lead to a non-analytic dependence of the impurity drift velocity on small forces.