Inorganic solid-state batteries are attracting significant interest as a contender to conventional liquid electrolyte-based lithium-ion batteries but still suffer from several limitations. The search for advanced coatings for protecting cathode materials in solid-state batteries to achieve interfacial stability is a continuing challenge. In the present work, the surface of an industrially relevant Ni-rich LiNi$_x$Co$_y$Mn$_z$O$_2$ cathode material, NCM-851005 (85 % Ni), was modified by applying a coating containing Li, Nb and Zn, aiming at a composition Li$_6$ZnNb$_4$O$_14$, by means of sol-gel chemistry. Detailed characterization using scanning transmission electron microscopy combined with energy-dispersive X-ray spectroscopy and nano-beam electron diffraction showed that the surface layer after heating in O$_2$ at 500 °C contains Li$_3$NbO$_4$ nanocrystals and Li$_2$CO$_3$, with Zn presumably acting as a dopant. The protective coating on the NCM-851005 secondary particles significantly increased the cycling performance (reversible capacity, rate capability etc.) and stability of full cells using argyrodite Li$_6$PS$_5$Cl as solid electrolyte. Interestingly, the level of improvement is superior to that achieved with conventional LiNbO$_3$ coatings.