Stochastic heating (SH) is a nonlinear heating mechanism driven by the violation of magnetic moment invariance due to large-amplitude turbulent fluctuations producing diffusion of ions toward higher kinetic energies in the direction perpendicular to the magnetic field. It is frequently invoked as a mechanism responsible for the heating of ions in the solar wind. Here, we quantify for the first time the proton SH rate Q⊥ at radial distances from the Sun as close as 0.16 au, using measurements from the first two Parker Solar Probe encounters. Our results for both the amplitude and radial trend of the heating rate, Q⊥ ∝ r−2.5, agree with previous results based on the Helios data set at heliocentric distances from 0.3 to 0.9 au. Also in agreement with previous results, Q⊥ is significantly larger in the fast solar wind than in the slow solar wind. We identify the tendency in fast solar wind for cuts of the core proton velocity distribution transverse to the magnetic field to exhibit a flattop shape. The observed distribution agrees with previous theoretical predictions for fast solar wind where SH is the dominant heating mechanism.