  Magnetometry measurements in high quality LiFeAs single-crystals reveal a
change in the sign of the magnetic hysteresis in the vicinity of the upper
critical field $H_{c2}$, from a clear diamagnetic response dominated by the
pinning of vortices, to a considerably smaller net hysteretic response of
opposite sign, which \emph{disappears} at $H_{c2}$. If the diamagnetic response
at high fields results from pinned vortices and associated screening
super-currents, this sign change must result from currents circulating in the
opposite sense, which give rise to a small field-dependent magnetic moment
\emph{below} $H_{c2}$. This behavior seems to be extremely sensitive to the
sample quality or stoichiometry, as we have observed it only in a few fresh
crystals, which also display the de Haas van Alphen-effect. We provide
arguments against the surface superconductivity, the flux compression, and the
random $\pi$ junction scenarios, which have been previously put forward to
explain a paramagnetic Meissner effect, below the lower critical field
$H_{c1}$. The observed anomalous hysteresis at high fields will be compatible
with the existence of chiral gap wave-functions, which possess a field
dependent magnetic moment. Within a Landau-Ginzburg framework, we demonstrate
how a $(d_{x^2 - y^2} + id_{xy})$ or a $(p_x+ip_y)$ chiral superconducting
component can be stabilized in the mixed state of $s_{\pm}$ superconductor, due
to the combined effects of the magnetic field and the presence of competing
pairing channels. The realization of a particular chiral pairing depends on the
microscopic details of the strengths of the competing pairing channels.
