  Dirac semimetals and Weyl semimetals are 3D analogs of graphene in which
crystalline symmetry protects the nodes against gap formation [1-3]. Na$_3$Bi
and Cd$_3$As$_2$ were predicted to be Dirac semimetals [4,5], and recently
confirmed to be so by photoemission [6-8]. Several novel transport properties
in a magnetic field $\bf H$ have been proposed for Dirac semimetals [2,9-11].
Here we report an interesting property in Cd$_3$As$_2$ that was unpredicted,
namely a remarkable protection mechanism that strongly suppresses
back-scattering in zero $\bf H$. In single crystals, the protection results in
a very high mobility that exceeds $>10^7$ cm$^2$/Vs below 4 K. Suppression of
backscattering results in a transport lifetime 10$^4\times$ longer than the
quantum lifetime. The lifting of this protection by $\bf H$ leads to an unusual
giant $\bf H$-linear magnetoresistance that violates Kohler's rule. We discuss
how this may relate to changes to the Fermi surface induced by $\bf H$.
