  We apply time-dependent R-matrix theory to study inner-shell ionization of C
atoms in ultra-short high-frequency light fields with a photon energy between
170 and 245 eV. At an intensity of 10$^{17}$ W/cm$^2$, ionization is dominated
by single-photon emission of a $2\ell$ electron, with two-photon emission of a
1s electron accounting for about 2-3\% of all emission processes, and
two-photon emission of $2\ell$ contributing about 0.5-1\%. Three-photon
emission of a 1s electron is estimated to contribute about 0.01-0.03\%. Around
a photon energy of 225 eV, two-photon emission of a 1s electron, leaving C$^+$
in either 1s2s2p$^3$ or 1s2p$^4$ is resonantly enhanced by intermediate
1s2s$^2$2p$^3$ states. The results demonstrate the capability of time-dependent
R-matrix theory to describe inner-shell ionization processes including
rearrangement of the outer electrons.
