  We study the zero-temperature ($T=0$) ground-state (GS) properties of a
frustrated spin-half $J^{XXZ}_{1}$--$J^{XXZ}_{2}$ model on the honeycomb
lattice with nearest-neighbor and next-nearest-neighbor interactions with
exchange couplings $J_{1}>0$ and $J_{2} \equiv \kappa J_{1}>0$, respectively,
using the coupled cluster method. Both interactions are of the anisotropic
$XXZ$ type. We present the $T=0$ GS phase diagram of the model in the ranges $0
\leq \Delta \leq 1$ of the spin-space anisotropy parameter and $0 \leq \kappa
\leq 1$ of the frustration parameter. A possible quantum spin-liquid region is
identified.
