A pulse-dilation photomultiplier tube (PD-PMT) with sub-20 ps temporal resolution has been developed for use with γ-ray-sensitive gas Cherenkov detectors at the National Ignition Facility to improve the diagnosis of nuclear fusion burn history and the areal density of the remaining capsule ablator. The pulse-dilation mechanism entails the application of a time-dependent, ramp waveform to a photocathode-mesh structure, introducing a time-dependent photoelectron accelerating potential. The electric field imparts axial velocity dispersion to outgoing photoelectrons. The photoelectron pulse is dilated as it transits a drift region prior to amplification in a microchannel plate and read out with a digital oscilloscope. We report the first measurements with the prototype PD-PMT demonstrating nominal <20 ps FWHM across a 400 ps measurement window and <30 ps FWHM for an extracted charge up to 300 pC. The output peak areas are linear to within 20% over 3 orders of magnitude of input intensity. 3D particle in cell simulations, which included space charge effects, have been carried out to investigate the device temporal magnification, resolution, and linearity.