We report in situ observations of an electron diffusion region (EDR) and adjacent separatrix region in the Earth's magnetotail. We observe significant magnetic field oscillations near the lower hybrid frequency which propagate perpendicularly to the reconnection plane. We also find that the strong electron-scale gradients close to the EDR exhibit significant oscillations at a similar frequency. Such oscillations are not expected for a crossing of a steady 2D EDR, and can be explained by a complex motion of the reconnection plane induced by current sheet kinking propagating in the out-of-reconnection-plane direction. Thus, all three spatial dimensions have to be taken into account to explain the observed perturbed EDR crossing. These results shed light on the interplay between magnetic reconnection and current sheet drift instabilities in electron-scale current sheets and highlight the need for adopting a 3D description of the EDR, going beyond the two-dimensional and steady-state conception of reconnection.