Epitaxial La2/3Sr1/3MnO3 (LSMO) films have been grown on SrTiO3 (001) substrates via pulsed laser deposition. In a 22-nm-thick LSMO film with a low residual resistivity of ρ0≈59μΩcm, 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 sub-band states play a crucial role for facilitating marked SOC in clean LSMO. Moreover, we find that the SOC strength and gate voltage tunability are similar to those in a two-dimensional electron gas at the LaAlO3/SrTiO3 interface, indicating the presence of an internal electric field near the LSMO/SrTiO3 interface. In a control measurement on a 5-nm-thick high resistivity (ρ0≈280μΩcm) LSMO film, we observe only a small zero-field peak in MR from weak localization effect, indicating negligible SOC.