One of the highlights of the European research infrastructure landscape is the world's most powerful X-ray-laser, the European XFEL. The ELBEX (Extracted Lepton Beam at the European XFEL) proposal builds on this strength and will set up new opportunities for European scientists and innovators, by providing an extracted high energy electron beam for experiments. With ELBEX we propose a pathfinder project to demonstrate the feasibility of such a facility at the European XFEL. This unique new possibility would strengthen the global competitiveness of the European Research Area and create opportunities for new user groups. The high energy, high charge density and excellent quality of the electron beam, if brought into interaction with a strong laser beam, opens up the study of a range of scientific topics, most prominently, of strong field Quantum Electrodynamics (QED). For the first time, the Schwinger limit for the electromagnetic field strength, at which non-perturbative QED effects become relevant, could be reached experimentally in this facility. Studying the particles created in the photon beam dump opens up the possibility to search for feebly interacting particles, complementing current or planned experiments like FASER II or SHiP. In addition, the electron beam itself is at the centre of a range of highly relevant and ambitious experiments in the area of accelerator science and detector science. Within the ELBEX project, the installation of a facility to extract an electron beam from the European XFEL using a fast kicker magnet and to transport it into a multi-purpose experimental area will be prepared. On the condition that a positive decision by the European XFEL council is reached to grant an extended 12-week XFEL shutdown, the installation of the ELBEX facility is an option.