Coherent extreme-ultraviolet radiation extending to wavelengths below the carbon K edge at 4.37 nanometers (nm) has been generated at a repetition rate of 1 kilohertz by focusing 5-femtosecond near-infrared (780 nm) laser pulses into a helium gas jet. The incident light field performs just a few oscillations, which results in the emission of an x-ray supercontinuum rather than discrete harmonics. Owing to the extremely short rise time of the driving pulses, neutral atoms can be exposed to high fields before they are depleted by ionization. As a result, the observed x-ray radiation extends well into the water window and is delivered in a well-collimated beam (divergence less than 1 milliradian). The high repetition rate and spatial coherence result in a brightness of about 5 × 10 ⁸ photons per square millimeter per square milliradian per second in a 1-percent bandwidth at 4.37 nm, the carbon edge of the water window. The compact laboratory system holds promise as a source for biological holography and nonlinear optics in the x-ray regime.