  A search for heavy Majorana neutrinos produced in the $B^- \to
\pi^+\mu^-\mu^-$ decay mode is performed using 3 fb$^{-1}$ of integrated
luminosity collected with the LHCb detector in $pp$ collisions at
center-of-mass energies of 7 TeV and 8 TeV at the LHC. Neutrinos with masses in
the range 250-5000 MeV and lifetimes from zero to 1000 ps are probed. In the
absence of a signal, upper limits are set on the branching fraction
${\cal{B}}(B^- \to \pi^+\mu^-\mu^-)$ as functions of neutrino mass and
lifetime. These limits are on the order of $10^{-9}$ for short neutrino
lifetimes of 1 ps or less. Limits are also set on the coupling between the muon
and a possible fourth-generation neutrino.
