Sequential tunneling of electrons through a single chlorine vacancy on a multilayer NaCl film on Cu(100) in the junction of a non-contact atomic force microscope with a conductive tip can lead to a negative damping of the cantilever. The characteristic features in the damping signal as a function of NaCl-layer thickness, tip position and tip-sample bias can be explained in a simple rate-equation model for the sequential tunneling process through the double-barrier tunnel junction. The first barrier results from the vacuum gap between tip and vacancy and its tunneling rate is tuned by the tip position and the applied bias. The second barrier results from the NaCl film and its tunneling rate is governed by the film's thickness. The current-induced damping is strongest if the two tunneling rates through each of the barriers individually are both comparable to the cantilever's oscillation frequency. The damping signal can be employed for detecting subfemtoampere tunneling currents and furthermore, maps of the damping signal can be used to inspect the mesoscopic tip shape. The observed and described current-induced damping should be a common phenomenon in non-contact atomic force microscopy for double-barrier tunnel junction geometries.