We analyze two high-resolution spectral transit time series of the hot Jupiter HAT-P-32 b obtained with the CARMENES spectrograph. Our new XMM-Newton X-ray observations of the system show that the fast-rotating F-type host star exhibits a high X-ray luminosity of 2.3 × 1029 erg s−1 (5-100 Å), corresponding to a flux of 6.9 × 104 erg cm−2 s−1 at the planetary orbit, which results in an energy-limited escape estimate of about 1013 g s−1 for the planetary mass-loss rate. The spectral time series show significant, time-dependent absorption in the Hα and He Iλ10833 triplet lines with maximum depths of about 3.3% and 5.3%. The mid-transit absorption signals in the Hα and He Iλ10833 lines are consistent with results from one-dimensional hydrodynamic modeling, which also yields mass-loss rates on the order of 1013 g s−1. We observe an early ingress of a redshifted component of the transmission signal, which extends into a redshifted absorption component, persisting until about the middle of the optical transit. While a super-rotating wind can explain redshifted ingress absorption, we find that an up-orbit stream, transporting planetary mass in the direction of the star, also provides a plausible explanation for the pre-transit signal. This makes HAT-P-32 a benchmark system for exploring atmospheric dynamics via transmission spectroscopy.