  We study the zero-temperature ground-state (GS) phase diagram of a spin-half
$J_{1}$--$J_{2}$ $XY$ model on the honeycomb lattice with nearest-neighbor
exchange coupling $J_{1}>0$ and frustrating next-nearest-neighbor exchange
coupling $J_{2} \equiv \kappa J_{1}>0$, where both bonds are of the isotropic
$XY$ type, using the coupled cluster method. Results are presented for the GS
energy per spin, magnetic order parameter, and staggered dimer valence-bond
crystalline (SDVBC) susceptibility, for values of the frustration parameter in
the range $0 \leq \kappa \leq 1$. In this range we find phases exhibiting,
respectively, N\'{e}el $xy$ planar [N(p)], N\'{e}el $z$-aligned [N($z$)],
SDVBC, and N\'{e}el-II $xy$ planar [N-II(p)] orderings which break the lattice
rotational symmetry. The N(p) state, which is stable for the classical version
of the model in the range $0 \leq \kappa \leq \frac{1}{6}$, is found to form
the GS phase out to a first quantum critical point at $\kappa_{c_{1}} =
0.216(5)$, beyond which the stable GS phase has N($z$) order over the range
$\kappa_{c_{1}} < \kappa < \kappa_{c_{2}}=0.355(5)$. For values $\kappa >
\kappa_{c_{2}}$ we find a strong competition to form the GS phase between
states with N-II(p) and SDVBC forms of order. Our best estimate, however, is
that the stable GS phase over the range $\kappa_{c_{2}} < \kappa <
\kappa_{c_{3}} \approx 0.52(3)$ is a mixed state with both SDVBC and N-II(p)
forms of order; and for values $\kappa > \kappa_{c_{3}}$ is the N-II(p) state,
which is stable at the classical level only at the highly degenerate point
$\kappa=\frac{1}{2}$. Over the range $0 \leq \kappa \leq 1$ we find no evidence
for any of the spiral phases that are present classically for all values
$\kappa > \frac{1}{6}$, nor for any quantum spin-liquid state.
