The detection of the first electromagnetic counterpart to the binary neutron star (BNS) merger remnant GW170817 established the connection between short gamma-ray bursts and BNS mergers. It also confirmed the forging of heavy elements in the ejecta (a so-called kilonova) via the r-process nucleosynthesis. The appearance of nonthermal radio and X-ray emission, as well as the brightening, which lasted more than 100 days, were somewhat unexpected. Current theoretical models attempt to explain this temporal behavior as either originating from a relativistic off-axis jet or a kilonova-like outflow. In either scenario, there is some ambiguity regarding how much energy is transported in the nonthermal electrons versus the magnetic field of the emission region. Combining the Very Large Array (radio) and Chandra (X-ray) measurements with observations in the GeV-TeV domain can help break this ambiguity, almost independently of the assumed origin of the emission. Here we report for the first time on deep H.E.S.S. observations of GW170817/GRB 170817A between 124 and 272 days after the BNS merger with the full H.E.S.S. array of telescopes, as well as on an updated analysis of the prompt (<5 days) observations with the upgraded H.E.S.S. phase-I telescopes. We discuss implications of the H.E.S.S. measurement for the magnetic field in the context of different source scenarios.