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Abstract. In this paper, we study the possibility to produce femtosecond monochromatic coherent ("x-ray laser") pulses in the keV range from inner-shell photo-ionisation of a gaseous medium by a laser-driven betatron pump source. A collisional-radiative model and a radiative transfer post-processor have been built to determine the lasing plasma kinetics, assess the local gain coefficient along the K transition in the ASE regime, and finally compute the x-ray laser intensity at saturation, to infer the key parameters for an experimental demonstration.