An elastic body moving in a quenched random force environment with a singular force correlator experiences an additional friction force which is inversely proportional to velocity. This dynamical friction mechanism is important, e.g., for a moving interface which is subjected to quenched random pinning forces. For sufficiently strong pinning forces, interface motion becomes unstable at values of driving force smaller than some critical one and the interface velocity drops. We expect in this case a first-order depinning transition with hysteresis in the dependence of the interface velocity on driving force. Instability in the interface motion remains under perturbations like an additional periodic driving force or thermal fluctuations. Interface motion instability can also take place in systems with small viscosity, when inertial effects are important.