  The transverse momentum ($p_{\rm T}$) differential production cross section
of the promptly-produced charm-strange baryon $\Xi_{\rm c}^{0}$ (and its charge
conjugate $\overline{\Xi_{\rm c}^{0}}$) is measured at midrapidity via its
hadronic decay into ${\rm \pi^{+}}\Xi^{-}$ in p$-$Pb collisions at a
centre-of-mass energy per nucleon$-$nucleon collision
$\sqrt{s_{\mathrm{NN}}}~=~5.02$ TeV with the ALICE detector at the LHC. The
$\Xi_{\rm c}^{0}$ nuclear modification factor ($R_{\rm pPb}$), calculated from
the cross sections in pp and p$-$Pb collisions, is presented and compared with
the $R_{\rm pPb}$ of $\Lambda_{\rm c}^{+}$ baryons. The ratios between the
$p_{\rm T}$-differential production cross section of $\Xi_{\rm c}^{0}$ baryons
and those of $\mathrm {D^0}$ mesons and $\Lambda_{\rm c}^{+}$ baryons are also
reported and compared with results at forward and backward rapidity from the
LHCb Collaboration. The measurements of the production cross section of prompt
$\Xi^0_{\rm c}$ baryons are compared with a model based on perturbative QCD
calculations of charm-quark production cross sections, which includes only cold
nuclear matter effects in p$-$Pb collisions, and underestimates the measurement
by a factor of about 50. This discrepancy is reduced when the data is compared
with a model in which hadronisation is implemented via quark coalescence. The
$p_{\rm T}$-integrated cross section of prompt $\Xi^0_{\rm c}$-baryon
production at midrapidity extrapolated down to $p_{\rm T}$ = 0 is also
reported. These measurements offer insights and constraints for theoretical
calculations of the hadronisation process. Additionally, they provide inputs
for the calculation of the charm production cross section in p$-$Pb collisions
at midrapidity.
