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They typically operate with low-energy, highly compressed laser pulses focused on high-pressure gas targets. Experiments have shown that the best-quality electron beams are achieved using hydrogen gas targets. However, continuous operation with hydrogen requires a dedicated pumping system. In this work, we present a method for designing a differential pumping system, which we successfully implemented in our experiments. This enabled the first demonstration of continuous operation of a kHz LWFA using a high-pressure hydrogen gas jet. The system effectively maintained a pressure below 3 \u00d7 10\u22124 mbar, even with a free-flowing gas jet operating at 140\u00a0bar backing pressure. 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