Context. Young massive stellar clusters are extreme environments and potentially provide the means for efficient particle acceleration. Indeed, they are increasingly considered as being responsible for a significant fraction of cosmic rays (CRs) that are accelerated within the Milky Way. Westerlund 1, the most massive known young stellar cluster in our Galaxy, is a prime candidate for studying this hypothesis. While the very-high-energy gamma-ray source HESS J1646-458 has been detected in the vicinity of Westerlund 1 in the past, its association could not be firmly identified. Aims. We aim to identify the physical processes responsible for the gamma-ray emission around Westerlund 1 and thus to understand the role of massive stellar clusters in the acceleration of Galactic CRs better. Methods. Using 164 h of data recorded with the High Energy Stereoscopic System (H.E.S.S.), we carried out a deep spectromorphological study of the gamma-ray emission of HESS J1646-458. We furthermore employed H I and CO observations of the region to infer the presence of gas that could serve as target material for interactions of accelerated CRs. Results. We detected large-scale (similar to 2 degrees diameter) gamma-ray emission with a complex morphology, exhibiting a shell-like structure and showing no significant variation with gamma-ray energy. The combined energy spectrum of the emission extends to several tens of TeV, and it is uniform across the entire source region. We did not find a clear correlation of the gamma-ray emission with gas clouds as identified through H I and CO observations. Conclusions. We conclude that, of the known objects within the region, only Westerlund 1 can explain the majority of the gamma-ray emission. Several CR acceleration sites and mechanisms are conceivable and discussed in detail. While it seems clear that Westerlund 1 acts as a powerful particle accelerator, no firm conclusions on the contribution of massive stellar clusters to the flux of Galactic CRs in general can be drawn at this point.