We prove that smooth critical points of the Möbius energy E parametrized by arc-length are analytic. Together with the main result in Blatt et al. (Trans Am Math Soc 368(9):63916438, 2016) this implies that critical points of the Möbius energy with merely bounded energy are not only C but also analytic. Our proof is based on Cauchys method of majorants and a decomposition of the gradient which has already proved useful in the proof of the regularity results in Blatt and Reiter (Manuscr Math 140(12):2950, 2013) and Blatt et al. (Trans Am Math Soc 368(9):63916438, 2016). To the best of the authors knowledge, this is the first analyticity result in the context of non-local differential equations.