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Ongoing progress in laser and accelerator technology opens new possibilities in high-field science, notably for the study of the largely unexplored strong-field QED regime where electron-positron pairs can be created directly from light-matter or even light-vacuum interactions. Laserless strategies such as beam-beam collisions have also been proposed with the prospect of pushing strong-field quantum electrodynamics (SFQED) in the nonpertubative regime. Here we report on an original concept to probe strong-field QED by harnessing the interaction between an electron beam and a solid target. When a high-density, ultrarelativistic beam impinges onto an even denser plasma, the beam self fields are reflected at the plasma boundary. In the rest frame of the beam electrons, these fields can exceed the Schwinger field, leading to strong-field QED effects such as quantum nonlinear inverse Compton scattering and nonlinear Breit-Wheeler electron-positron pair creation. We show that such beam-plasma collisions can produce results similar to beam-beam collisions with the advantage of a much simpler experimental setup including the automatic overlap between the beam and the reflected fields. This scenario opens the way to precision studies of strong-field QED, with measurable clear signatures in terms of gamma-ray photon and pair production, and thus is a very promising milestone on the path towards laserless studies of nonperturbative SFQED.