  This article presents the first measurement of the differential $Z$-boson
production cross-section in the forward region using proton-lead collisions
with the LHCb detector. The dataset was collected at a nucleon-nucleon
centre-of-mass energy of $\sqrt{s_\mathrm{NN}}=8.16\,\mathrm{TeV}$ in 2016,
corresponding to an integrated luminosity of $30.8\,\mathrm{nb}^{-1}$. The
forward-backward ratio and the nuclear modification factors are measured
together with the differential cross-section as functions of the $Z$ boson
rapidity in the centre-of-mass frame, the transverse momentum of the $Z$ boson
and a geometric variable $\phi^{*}$. The results are in good agreement with the
predictions from nuclear parton distribution functions, providing strong
constraining power at small Bjorken-$x$.
