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) 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. We aim to identify the physical processes responsible for the $\gamma$-ray emission around Westerlund 1 and thus to better understand the role of massive stellar clusters in the acceleration of Galactic CRs. Using 164 hours 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. We detected large-scale ($\sim 2^\circ$ 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 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. We conclude that, of the known objects within the region, only Westerlund 1 can explain the bulk of the $\gamma$-ray emission. Several CR acceleration sites and mechanisms are conceivable, and discussed in detail. (abridged)< Leer menos