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In this paper, we study the quasinormal modes of the massless Dirac field for charged black holes in Rastall gravity. The spherically symmetric black hole solutions in question are characterized by the presence of a power-Maxwell field, surrounded by the quintessence fluid. The calculations are carried out by employing the WKB approximations up to the thirteenth order, as well as the matrix method. The temporal evolution of the quasinormal modes is investigated by using the finite difference method. Through numerical simulations, the properties of the quasinormal frequencies are analyzed, including those for the extremal black holes. Among others, we explore the case of a second type of extremal black holes regarding the Nariai solution, where the cosmical and event horizon coincide. The results obtained by the WKB approaches are found to be mostly consistent with those by the matrix method. It is demonstrated that the black hole solutions for Rastall gravity in asymptotically flat spacetimes are equivalent to those in Einstein gravity, featured by different asymptotical spacetime properties. As one of its possible consequences, we also investigate the behavior of the late-time tails of quasinormal models in the present model. It is found that the asymptotical behavior of the late-time tails of quasinormal modes in Rastall theory is governed by the asymptotical properties of the spacetimes of their counterparts in Einstein gravity.