Mixtures of ultracold gases with long-range interactions are expected to open new avenues in the study of quantum matter. Natural candidates for this research are spin mixtures of atomic species with large magnetic moments. However, the lifetime of such assemblies can be strongly affected by the dipolar relaxation that occurs in spin-flip collisions. Here we present experimental results for a mixture composed of the two lowest Zeeman states of $^{162}$Dy atoms, that act as dark states with respect to a light-induced quadratic Zeeman effect. We show that, due to an interference phenomenon, the rate for such inelastic processes is dramatically reduced with respect to the Wigner threshold law. Additionally, we determine the scattering lengths characterizing the s-wave interaction between these states, providing all necessary data to predict the miscibility range of the mixture, depending on its dimensionality.