In this paper, we present an open-shell extension of the non-orthogonal configuration interaction singles (NOCIS) method for the calculation of core-excited states, intended for peak assignment in XAS spectra of doublet radicals. This extension requires the consideration of additional configurations due to the singly occupied open-shell orbital, and the addition of essential orbital relaxation effects is found to provide a significant improvement on standard CIS, while maintaining the desirable properties of spin purity, variationality, and size consistency. We apply this method to the calculation of core excitations for several open-shell molecules and demonstrate that it performs competitively with other available methods, despite a lack of dynamic correlation. In particular, relative to CVS-ADC(2)-x, RMS error is reduced by a factor of 6 over usual orthogonal CIS and is comparable to time-dependent density functional theory with the best short-range corrected functionals
