  We have performed an angle-resolved photoemission spectroscopy study of the
new superconductor Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ in the low energy range. We
report the observation of an anomaly around 25 meV in the dispersion of
superconducting Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ samples that nearly vanishes
above $T_c$. The energy scale of the related mode (13$\pm$2 meV) and its strong
dependence on orbital and temperature indicates that it is unlikely related to
phonons. Moreover, the momentum locations of the kink can be connected by the
antiferromagnetic wavevector. Our results point towards an unconventional
electronic origin of the mode and the superconducting pairing in the Fe-based
superconductors, and strongly support the anti-phase s-wave pairing symmetry.
