  We consider the Fermi polaron problem of an impurity hopping around a
two-dimensional square lattice and interacting with a sea of fermions at given
filling factor. When the interaction is attractive, we find standard Fermi
polaron quasiparticles, categorized as attractive polarons and repulsive
polarons. When the interaction becomes repulsive, interestingly, we observe an
unconventional highly-excited polaron quasiparticle, sharply peaked at the
corner of the first Brillouin zone with momentum \mathbf{k}=(\pm\pi,\pm\pi).
This super Fermi polaron branch arises from the dressing of the impurity's
motion with holes, instead of particles of fermions. We show that super Fermi
polarons become increasingly well-defined with increasing impurity-fermion
repulsions and might be considered as a precursor of Nagaoka ferromagnetism,
which would appear at sufficiently large repulsions and at large filling
factors. We also investigate the temperature-dependence of super Fermi polarons
and find that they are thermally robust against the significant increase in
temperature.
