  We perform a calculation of inclusive $Z$ boson production in proton-lead
collisions at the LHC taking into account the transverse momenta of the initial
partons. We use the framework of $k_T$-factorization combining transverse
momentum dependent parton distributions (TMDs) with off-shell matrix elements.
In order to do it we need to construct appropriate TMDs for lead nuclei which
is done using the parton branching method. Our computations are compared with
data from CMS taken at $\sqrt{s}=5.02$ TeV. The results are in good agreement
with the measurements especially the transverse momentum distribution of the
$Z$ boson.
