  Strongly interacting fermionic systems host a variety of interesting quantum
many-body states with exotic excitations. For instance, the interplay of strong
interactions and the Pauli exclusion principle can lead to Stoner
ferromagnetism, but the fate of this state remains unclear when kinetic terms
are added. While in many lattice models the fermions' dispersion results in
delocalization and destabilization of the ferromagnet, flat bands can restore
strong interaction effects and ferromagnetic correlations. To reveal this
interplay, here we propose to study the Hofstadter-Fermi-Hubbard model using
ultracold atoms. We demonstrate, by performing large-scale DMRG simulations,
that this model exhibits a lattice analog of the quantum Hall ferromagnet at
magnetic filling factor $\nu=1$. We reveal the nature of the low energy
spin-singlet states around $\nu\approx1$ and find that they host
quasi-particles and quasi-holes exhibiting spin-spin correlations reminiscent
of skyrmions. Finally, we predict the breakdown of flat-band ferromagnetism at
large fields. Our work paves the way towards experimental studies of lattice
quantum Hall ferromagnetism, including prospects to study many-body states of
interacting skyrmions and explore the relation to high-$T_{\rm c}$
superconductivity.
