  Transition frequencies and fine-structure splittings of the $2\,{}^3\!S_1
\rightarrow 2\,{}^3\!P_{\!J}$ transitions in helium-like $^{12}\mathrm{C}^{4+}$
were measured by collinear laser spectroscopy on a 1-ppb level. Accuracy is
increased by more than three orders of magnitude with respect to previous
measurements, enabling tests of recent non-relativistic QED calculations
including terms up to $m\alpha^7$. Deviations between the theoretical and
experimental values are within theoretical uncertainties and are ascribed to
$m\alpha^8$ and higher-order contributions in the series expansion of the
NR-QED calculations. Finally, prospects for an all-optical charge radius
determination of light isotopes are evaluated.
