  Using the Hydro-Coal-Frag model that combines hydrodynamics at low $p_{\rm
T}$, quark coalescence at intermediate $p_{\rm T}$, and the LBT transport model
at high $p_{\rm T}$, we study the spectra and elliptic flow of identified
hadrons in high multiplicity p--Pb and p--p collisions at the Large Hadron
Collider (LHC). In p--Pb collisions, the Hydro-Coal-Frag model gives a good
description of the differential elliptic flow over the $p_{\rm T}$ range from 0
to 6 GeV and the approximate number of constituent quark (NCQ) scaling at
intermediate $p_{\rm T}$. Although Hydro-Coal-Frag model can also roughly
describe the elliptic flow in high multiplicity p--p collisions with the quark
coalescence process, the larger contribution from the string fragmentations
leads to a notable violation of the NCQ scaling of $v_2$ at intermediate
$p_{\rm T}$ as observed in the experiment. Comparison runs of the Hydro-Frag
model without the coalescence process demonstrate that regardless the parameter
adjustments, the Hydro-Frag model cannot simultaneously describe the $p_{\rm
T}$ spectra and the elliptic flow of identified hadrons in either p--Pb
collisions or p--p collisions. The calculations in this paper thus provide
support for the existence of partonic degrees of freedom and the possible
formation of the QGP in the small systems created at the LHC.
