  Epitaxial La$_{2/3}$Sr$_{1/3}$MnO$_3$ (LSMO) films have been grown on
SrTiO$_3$ (001) substrates via pulsed laser deposition. In a 22-nm thick LSMO
film with a low residual resistivity of $\rho_0$ = 59 $\mu \Omega$ cm, we found
a zero-field dip in the magnetoresistance (MR) below 10 K, manifesting the weak
antilocalization (WAL) effect due to strong spin-orbit coupling (SOC). We have
analyzed the MR data by including the D'yakonov-Perel' spin-relaxation
mechanism in the WAL theory. We explain that the delocalized spin-down electron
subband states play a crucial role for facilitating marked SOC in clean LSMO.
Moreover, we find that the SOC strength and gate voltage tunability is similar
to that in the 2DEG at LaAlO$_3$/SrTiO$_3$ interface, indicating the presence
of an internal electric field near the LSMO/SrTiO$_3$ interface. In a control
measurement on a 5-nm thick high resistivity ($\rho_0$ = 280 $\mu \Omega$ cm)
LSMO film, we observe only a small zero-field peak in MR from weak localization
effect, indicating negligible SOC.
