Dark matter particles could be superheavy, provided their lifetime is much longer than the age of the universe. Using the sensitivity of the Pierre Auger Observatory to ultra-high energy neutrinos and photons, we constrain a specific extension of the Standard Model of particle physics that meets the lifetime requirement for a superheavy particle by coupling it to a sector of ultra-light sterile neutrinos. Our results show that, for a typical dark coupling constant of 0.1, the mixing angle $\theta_m$ between active and sterile neutrinos must satisfy, roughly, $\theta_m \lesssim 1.5\times 10^{-6}(M_X/10^9~\mathrm{GeV})^{-2}$ for a mass $M_X$ of the dark-matter particle between $10^8$ and $10^{11}~$GeV.