The formation and stability of collisionless self-gravitating systems are long-standing problems, which date back to the work of D. Lynden-Bell on violent relaxation and extends to the issue of virialization of dark matter (DM) haloes. An important prediction of such a relaxation process is that spherical equilibrium states can be described by a Fermi–Dirac phase-space distribution, when the extremization of a coarse-grained entropy is reached. In the case of DM fermions, the most general solution develops a degenerate compact core surrounded by a diluted halo. As shown recently, the latter is able to explain the galaxy rotation curves, while the DM core can mimic the central black hole. A yet open problem is whether these kinds of astrophysical core–halo configurations can form at all, and whether they remain stable within cosmological time-scales. We assess these issues by performing a thermodynamic stability analysis in the microcanonical ensemble for solutions with a given particle number at halo virialization in a cosmological framework. For the first time, we demonstrate that the above core–halo DM profiles are stable (i.e. maxima of entropy) and extremely long-lived. We find the existence of a critical point at the onset of instability of the core–halo solutions, where the fermion-core collapses towards a supermassive black hole. For particle masses in the keV range, the core-collapse can only occur for |$M_{\rm vir} \gtrsim 10^{9}{\, \mathrm{M}_\odot}$| starting at z_vir ≈ 10 in the given cosmological framework. Our results prove that DM haloes with a core–halo morphology are a very plausible outcome within non-linear stages of structure formation.