It is shown that, for a class of Hamiltonians of XXZ chains in an external, longitudinal magnetic field that are small perturbations of an Ising Hamiltonian, the spectral gap above the ground-state energy remains strictly positive when the perturbation is turned on, uniformly in the length of the chain. The result is proven for both the ferromagnetic and the antiferromagnetic Ising Hamiltonian; in the latter case the external magnetic field is required to be small and the two-fold degenerate ground-state energy of the unperturbed Hamiltonian may split into two energy levels whose difference is bounded above by a fractional power of the (small) coupling constant of the transverse terms. This result is proven by using a new, quite subtle refinement of a method developed in earlier work and used to iteratively block-diagonalize Hamiltonians of ever larger subsystems with the help of local unitary conjugations. One novel ingredient of the method presented in this paper consists of the use of Lieb-Robinson bounds.
