Collisions induced by (9,10,11)Be on a 64Zn target at the same c.m. energy were studied. For the first time, strong effects of the 11Be halo structure on elastic-scattering and reaction mechanisms at energies near the Coulomb barrier are evidenced experimentally. The elastic-scattering cross section of the 11Be halo nucleus shows unusual behavior in the Coulomb-nuclear interference peak angular region. The extracted total-reaction cross section for the 11Be collision is more than double the ones measured in the collisions induced by (9,10)Be. It is shown that such a strong enhancement of the total-reaction cross section with 11Be is due to transfer and breakup processes.