Similar to electron waves, the phonon states in semiconductors can undergo changes induced by external boundaries. However, despite strong scientific and practical importance, conclusive experimental evidence of confined acoustic phonon polarization branches in individual free-standing nanostructures is lacking. Here we report results of Brillouin-Mandelstam light scattering spectroscopy, which reveal multiple (up to ten) confined acoustic phonon polarization branches in GaAs nanowires with a diameter as large as 128 nm, at a length scale that exceeds the grey phonon mean-free path in this material by almost an order-of-magnitude. The dispersion modification and energy scaling with diameter in individual nanowires are in excellent agreement with theory. The phonon confinement effects result in a decrease in the phonon group velocity along the nanowire axis and changes in the phonon density of states. The obtained results can lead to more efficient nanoscale control of acoustic phonons, with benefits for nanoelectronic, thermoelectric and spintronic devices.