  We systematically investigate the scaling property of mesons (pions and
kaons) and baryons (protons, ${\rm \Lambda}$, ${\rm \Xi}$ and ${\rm \Omega}$)
transverse momentum ($p_{\rm T}$) spectra at different centrality classes
(0-5$\%$, 5-10$\%$, 10-20$\%$, 20-40$\%$, 40-60$\%$, 60-80$\%$ and 80-100$\%$)
in proton-lead collisions with center of mass energy per nucleon pair 5.02 TeV.
In the low $p_{\rm T}$ region with $p_{\rm{T}}\leq $ 3.9 (3.1, 2.5, 2.7, 2.4
and 2.8) GeV/c, a universal scaling independent of the centrality is observed
in the pion (kaon, proton, ${\rm \Lambda}$, ${\rm \Xi}$ and ${\rm \Omega}$)
spectra when a dilatation, $p_{\rm T}\rightarrow p_{\rm T}/K$, is applied. Here
$K$ is a scaling parameter depending on the centrality class. We find that the
rates at which ln$K$ changes with the logarithmic value of the average value of
the number of participating nucleons, ln$\langle N_{\rm{part}}\rangle$, are
stronger for baryons than those for mesons. In the high $p_{\rm T}$ region,
there is a deviation from the scaling. The more peripheral the collisions are,
the more obvious the violation of the scaling is. In the framework of the
colour string percolation (CSP) model, we show that mesons and baryons are
generated from the decay of clusters formed by strings overlapping in the
transverse plane with the same size dispersion but with different mean size.
The mean size of clusters for baryons is smaller than that of mesons. For the
same hadrons at different centrality classes, the mean size of clusters
decreases with the increase of centrality. The fragmentation functions for
cluster decay are different for different hadrons, while they are universal for
the same hadrons at different centrality classes. The universal scaling of the
meson and baryon spectra in the low $p_{\rm T}$ region can be quantitatively
understood with the CSP model at the same time.
