Angular distributions of the elastic, inelastic, and breakup cross sections of the halo nucleus ^{11}Be on ^{197}Au were measured at energies below (E_{lab}=31.9 MeV) and around (39.6 MeV) the Coulomb barrier. These three channels were unambiguously separated for the first time for reactions of ^{11}Be on a high-Z target at low energies. The experiment was performed at TRIUMF (Vancouver, Canada). The differential cross sections were compared with three different calculations: semiclassical, inert-core continuum-coupled-channels and continuum-coupled-channels ones with including core deformation. These results show conclusively that the elastic and inelastic differential cross sections can only be accounted for if core-excited admixtures are taken into account. The cross sections for these channels strongly depend on the B(E1) distribution in ^{11}Be, and the reaction mechanism is sensitive to the entanglement of core and halo degrees of freedom in ^{11}Be.