  If QGP-like medium is created in $p$-Pb collisions at extremely high
collision energies, charm quarks that move in the medium can hadronize by
capturing the co-moving light quark(s) or anti-quark(s) to form the charm
hadrons. Using light quark $p_{T}$ spectra extracted from the experimental data
of light-flavor hadrons and a charm quark $p_{T}$ spectrum that is consistent
with perturbative QCD calculations, the central-rapidity data of $p_{T}$
spectra and the spectrum ratios for $D$ mesons in the low $p_{T}$ range
($p_{T}\lesssim7$ GeV/$c$) in minimum-bias $p$-Pb collisions at
$\sqrt{s_{NN}}=5.02$ TeV are well described by quark combination mechanism in
equal-velocity combination approximation. The $\Lambda_{c}^{+}/D^{0}$ ratio in
quark combination mechanism exhibits the typical increase-peak-decrease
behavior as the function of $p_{T}$, and the shape of the ratio for
$p_{T}\gtrsim3$ GeV/$c$ is in agreement with the data of ALICE collaboration in
central rapidity region $-0.96<y<0.04$ and the preliminary data of LHCb
collaboration in forward rapidity region $1.5<y<4.0$. The global production of
single-charm baryons is quantified using the data and the possible enhancement
(relative to light flavor baryons) is discussed. The $p_{T}$ spectra of
$\Xi_{c}^{0}$, $\Omega_{c}^{0}$ in minimum-bias events and those of
single-charm hadrons in high-multiplicity event classes are predicted, which
serves as the further test of the possible change of the hadronization
characteristic for low $p_{T}$ charm quarks in the small system created in
$p$-Pb collisions at LHC energies.
