  Total and fiducial top pair ($t\bar{t}$) production cross sections in
proton-lead (pPb) collisions at $\sqrt{s_{NN}}$ = 8.16 TeV are computed at
next-to-next-to-leading-order (NNLO) accuracy including
next-to-next-to-leading-log (NNLL) gluon resummation, using the CT14 and CT10
proton parton distribution functions (PDF), and the EPPS16 and EPS09 nuclear
PDF parametrizations for the lead ion. The total cross sections amount to
$\sigma(pPb\to t\bar{t}+X) = 59.0 \pm
5.3$(CT14+EPPS16)$\,^{+1.6}_{-2.1}$(scale) nb, and $57.5 \pm
\,^{+4.3}_{-3.3}$(CT10+EPS09)$\,^{+1.5}_{-2.0}$(scale) nb, with small
modifications with respect to the result computed using the free proton PDF
alone. The normalized ratio of pPb to pp cross sections (nuclear modification
factor) is $R_{pPb} = 1.04 \,^{\pm 0.07(EPPS16)}_{\pm0.03(EPS09)}$. In the
lepton+jets decay mode, $t\bar{t} \to b\bar{b} W(\ell\nu) W(q\bar{q}')$, one
expects $\sim$600 $t\bar{t}$ events in the 180 nb$^{-1}$ integrated luminosity
collected in pPb collisions at the LHC so far, after typical acceptance and
efficiency losses. Differential cross sections at NLO accuracy are presented as
a function of transverse momentum and rapidity of the top quarks, and of their
decay b-jets and isolated leptons.
