The Koopman operator is a novel approach to embed nonlinear dynamics into linear models. This work shows its successful application for fault detection to a large-scale facility with challenging real-time requirements: the European XFEL, which is the worldwide largest linear particle accelerator. We concentrate on the superconducting radio-frequency cavities, from which 808 exist and whose effective operation directly influences the performance of the whole facility. Thus, a proper fault detection scheme is desired. While a nonlinear state-space description of the cavity dynamics is well-known, its usage along with an unscented Kalman filter is not able to cope with the challenging online implementation requirements. Therefore, in this paper, we apply the Koopman operator technique to identify a finite-dimensional linear approximation of the nonlinear system. For the data-driven identification, the model knowledge is exploited by choosing physically motivated basis functions. With the linear approximation at hand, a linear Kalman filter can be applied. Results are presented for real experimental data. Compared to the unscented Kalman filter, the same detection capability but a speed-up of three orders of magnitude in calculation time can be achieved with the Koopman-based Kalman filter, which enables its implementation to the real facility.