In this article, we present new results of high-order coupled cluster method (CCM) calculations, based on a Néel model state with spins aligned in the z-direction, for both the ground- and excited-state properties of the spin-half XXZ model on the linear chain, the square lattice, and the simple cubic lattice. In particular, the high-order CCM formalism is extended to treat the excited states of lattice quantum spin systems for the first time. Completely new results for the excitation energy gap of the spin-half XXZ model for these lattices are thus determined. These high-order calculations are based on a localized approximation scheme called the LSUBm scheme in which we retain all k-body correlations defined on all possible locales of m adjacent lattice sites (k≤m). The `raw' CCM LSUBm results are seen to provide very good results for the ground-state energy, sublattice magnetization, and the value of the lowest-lying excitation energy for each of these systems. However, in order to obtain even better results, two types of scheme for extrapolating the LSUBm results to the limit m→∞ (i.e., the exact solution in the thermodynamic limit) are presented. The extrapolated results provide extremely accurate results for the ground- and excited-state properties of these systems across a wide range of values of the anisotropy parameter.