We revisit the problem of realizing perfectly isotropic antennas based on recent developments in nonlocal antenna theory. The key obstacle against the existence of exactly isotropic antennas is traced back to the hairy ball theorem in algebraic and differential topology. A path blocking the applicability of this no-go theorem is to allow for complementary longitudinal modes to coexist with matching transverse modes in the antenna exterior region. Nonlocal antennas (radiators embedded into nonlocal metamaterial exterior domains) are possible future antenna technologies capable of dealing with both transverse and longitudinal modes. It is rigorously demonstrated that the recently proposed nonlocal Proca metamaterial (massive electromagnetism) leads to small nonlocal dipole antennas with perfectly isotropic radiation pattern. A tentative proposal for a possible experimental realization of perfectly isotropic radiators in the future is briefly outlined.