  We prove that smooth critical points of the M\"obius energy parametrized by
arc-length are analytic. Together with the main result in \cite{BRS16} this
implies that critical points of the M\"obius energy with merely bounded energy
are not only $C^\infty$ but also analytic. Our proof is based on Cauchy's
method of majorants and a decomposition of the gradient which already proved
useful in the proof of the regularity results in \cite{BR13} and \cite{BRS16}.
To best of the authors knowledge, this is the first analyticity result in the
context of non-local differential equations.
