The ASTRI Mini-Array is a next-generation system of nine imaging atmospheric Cherenkov telescopes that is going to be built at the Observatorio del Teide site. After a first phase, in which the instrument will be operated as an experiment prioritizing a schedule of primary science cases, an observatory phase is foreseen in which other significant targets will be pointed. We focus on the observational feasibility of extragalactic sources and on astrophysical processes that best complement and expand the ASTRI Mini-Array core science, presenting the most relevant examples that are at reach of detection over long-term time scales and whose observation can provide breakthrough achievements in the very-high energy extragalactic science. Such examples cover a wide range of gamma-ray emitters, including the study of AGN low states in the multi-TeV energy range, the possible detection of Seyfert galaxies with long exposures and the searches of dark matter lines above 10 TeV. Simulations of the presented objects show that the instrument performance will be competitive at multi-TeV energies with respect to current arrays of Cherenkov telescopes.
