We demonstrate the operation of a plasma-based extreme ultraviolet (XUV) laser at a 1 kHz repetition rate driven by infrared pump pulses of less than 20 mJ. The 41.8 nm laser pulses were generated in a Xe plasma created by optical-field ionization by the L1 Allegra laser at ELI Beamlines. The output power of the XUV laser lies in the few microwatt range, and the energy efficiency of this pumping scheme opens the way for further scaling in repetition rate and average power.