  The first observation of $Z$ boson production in proton-lead collisions at a
centre-of-mass energy per proton-nucleon pair of $\sqrt{s_{NN}}=5~\text{TeV}$
is presented. The data sample corresponds to an integrated luminosity of
$1.6~\text{nb}^{-1}$ collected with the LHCb detector. The $Z$ candidates are
reconstructed from pairs of oppositely charged muons with pseudorapidities
between 2.0 and 4.5 and transverse momenta above $20~\text{GeV}/c$. The
invariant dimuon mass is restricted to the range $60-120~\text{GeV}/c^2$. The
$Z$ production cross-section is measured to be
  \begin{eqnarray*}
  \sigma_{Z\to\mu^+\mu^-}(\text{fwd})&=&13.5^{+5.4}_{-4.0}\text{(stat.)}\pm1.2\text{(syst.)}~\text{nb}
\end{eqnarray*} in the direction of the proton beam and
  \begin{eqnarray*}
  \sigma_{Z\to\mu^+\mu^-}(\text{bwd}) &
=&10.7^{+8.4}_{-5.1}\text{(stat.)}\pm1.0\text{(syst.)}~\text{nb}
  \end{eqnarray*} in the direction of the lead beam, where the first
uncertainty is statistical and the second systematic.
