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-$x$.