  We explore with exact diagonalization the propagation of a single hole in
four magnetic phases of the t-J-like Kitaev-Heisenberg model on a honeycomb
lattice: the Neel antiferromagnetic, stripe, zigzag and Kitaev spin-liquid
phase. We find coherent propagation of spin-polaron quasiparticles in the
antiferromagnetic phase by a similar mechanism as in the $t$-$J$ model for
high-$T_c$ cuprates. In the stripe and zigzag phases clear quasiparticles
features appear in spectral functions of those propagators where holes are
created and annihilated on one sublattice, while they remain largely {\it
hidden} in those spectral functions that correspond to photoemission
experiments. As the most surprising result, we find a totally incoherent
spectral weight distribution for the spectral function of a hole moving in the
Kitaev spin-liquid phase in the strong coupling regime relevant for iridates.
At intermediate coupling the finite systems calculation reveals a well defined
quasiparticle at the $\Gamma$ point, however, we find that the gapless spin
excitations wipe out quasiparticles at finite momenta. Also for this more
subtle case we conclude that in the thermodynamic limit the lightly doped
Kitaev liquid phase does not support quasiparticle states in the neighborhood
of $\Gamma$, and therefore yields a {\it non-Fermi liquid}, contrary to earlier
suggestions based on slave-boson studies. These observations are supported by
the presented study of the dynamic spin-structure factor for the Kitaev spin
liquid regime.
