The problem of internal oscillations of kink-type solitons in a one-dimensional easy-plane antiferromagnet is studied by analytical methods. Apart from the Goldstone mode a second local mode which is due to the coupling between in-plane and out-of-plane spin components has been derived. For typical experimental conditions in a model system like (TMMC) the frequency of this mode is very close to the bottom of the magnon band. Considering both a twofold and a weak sixfold in-plane crystal field anisotropy we found that close to the spin-flop transition the separation of this local mode from the magnon band is increased and dramatically affected by the sixfold anisotropy. We show that quantum properties of this mode are not important.