Microquasars have emerged as promising candidates to explain the cosmic-ray flux at PeV energies. LHAASO observations reveal V4641 Sgr as the most extreme example so far, with a gamma-ray spectrum extending up to 800~TeV, requiring multi-PeV energy particles. The TeV emission is highly extended, challenging expectations given the reported low-inclination angle of the V4641 Sgr jets. We aim to spatially and spectrally resolve the gamma-ray emission from V4641 Sgr and investigate particle acceleration in the system. Using ~100h of H.E.S.S. data we perform a spectro-morphological study of the gamma-ray emission around V4641 Sgr. We employ HI and dedicated CO observations of the region to infer the presence of target material for cosmic-ray interactions. Multi-TeV emission around V4641 Sgr is detected with high significance. The emission region is elongated with major and minor axes: 0.34{deg}+/-0.04_stat_+/-0.01_syst_ and 0.06{deg}+/-0.01_stat_+/-0.01_syst_, respectively. A power-law spectrum with index ~1.8 is found and, together with results by other gamma-ray instruments, reveals for the first time a spectral energy distribution (SED) peaking at energies of ~100TeV. We find indications (3{sigma}) of a two-component morphology, with indistinguishable spectral properties. The position of V4641 Sgr is inconsistent with both the best-fit position of the single-component model and the dip between the two components. We find no significant evidence of energy-dependent morphology. No dense gas is found at any distance towards V4641 Sgr, placing an upper limit of n_gas_<~0.2cm^-3^ within the gamma-ray emission region. The peak of the SED at ~100TeV identifies V4641 Sgr as a candidate cosmic-ray accelerator beyond the Knee. However, the absence of dense target gas places stringent energetic constraints on hadronic interpretations. The H.E.S.S. measurement requires an unusually hard (~1.5) spectral index for protons. A leptonic scenario faces fewer obstacles, provided particle transport is fast enough to avoid losses and reproduce the observed energy-independent morphology, though the absence of bright X-ray emission across the gamma-ray emission region requires a magnetic field strength <~3muG. Our findings favour a leptonic origin of the gamma-ray emission. This conclusion does not exclude hadron acceleration in the V4641 Sgr system.