  The production of $J/\psi$ mesons with rapidity $1.5<y<4.0$ or $-5.0<y<-2.5$
and transverse momentum $p_\mathrm{T}<14 \mathrm{GeV}/c$ is studied with the
LHCb detector in proton-lead collisions at a nucleon-nucleon centre-of-mass
energy $\sqrt{s_{NN}}=5 \mathrm{TeV}$. The analysis is based on a data sample
corresponding to an integrated luminosity of about $1.6 \mathrm{nb}^{-1}$. For
the first time the nuclear modification factor and forward-backward production
ratio are determined separately for prompt $J/\psi$ mesons and $J/\psi$ from
$b$-hadron decays. Clear suppression of prompt $J/\psi$ production with respect
to proton-proton collisions at large rapidity is observed, while the production
of $J/\psi$ from $b$-hadron decays is less suppressed. These results show good
agreement with available theoretical predictions. The measurement shows that
cold nuclear matter effects are important for interpretations of the related
quark-gluon plasma signatures in heavy-ion collisions.
