  Carrier scattering processes are studied in CH$_3$NH$_3$PbI$_3$ using
temperature-dependent four-wave mixing experiments. Our results indicate that
scattering by ionized impurities limits the interband dephasing time (T$_2$)
below 30~K, with strong electron-phonon scattering dominating at higher
temperatures (with a timescale of 125 fs at 100 K). Our theoretical simulations
provide quantitative agreement with the measured carrier scattering rate and
show that the rate of acoustic phonon scattering is enhanced by strong
spin-orbit coupling, which modifies the band-edge density of states. The Rashba
coefficient extracted from fitting the experimental results ($\gamma_c=2$ eV
angstrom) is in agreement with calculations of the surface Rashba effect and
recent experiments using the photogalvanic effect on thin films.
