We observe the suppression of inelastic dipolar scattering due to Fermi statistics in ultracold gases of the highly magnetic atom, dysprosium. Inelastic dipolar scattering in non-zero magnetic fields leads to heating or to loss of the trapped population, both detrimental to experiments intended to study quantum many-body physics with strongly dipolar gases. Fermi statistics, however, should lead to a kinematic suppression of these harmful reactions. Indeed, we observe a 120-fold suppression of dipolar relaxation in fermionic versus bosonic Dy, as expected from theory describing universal inelastic dipolar scattering. Similarly low inelastic cross sections are observed in spin mixtures, also with striking correspondence to theory predictions. The suppression of relaxation opens the possibility of employing fermionic dipolar species---atoms or molecules---in studies of quantum many-body physics involving, e.g., synthetic gauge fields and pairing. See Physical Review Letters paper.