  Giant Gilbert damping anisotropy is identified as a signature of strong
Rashba spin-orbit coupling in a two-dimensional antiferromagnet on a honeycomb
lattice. The phenomenon originates in spin-orbit induced splitting of
conduction electron subbands that strongly suppresses certain spin-flip
processes. As a result, the spin-orbit interaction is shown to support an
undamped non-equilibrium dynamical mode that corresponds to an ultrafast
in-plane N\'eel vector precession and a constant perpendicular-to-the-plane
magnetization. The phenomenon is illustrated on the basis of a two dimensional
$s$-$d$ like model. Spin-orbit torques and conductivity are also computed
microscopically for this model. Unlike Gilbert damping these quantities are
shown to reveal only a weak anisotropy that is limited to the semiconductor
regime corresponding to the Fermi energy staying in a close vicinity of
antiferromagnetic gap.
