Context. Vela X is a pulsar wind nebula in which two relativistic particle populations with distinct spatial and spectral distributions dominate the emission at different wavelengths. An extended 2° × 3° nebula is seen in radio and GeV gamma rays. An elongated cocoon prevails in X-rays and TeV gamma rays. Aims. We use ~9.5 yr of data from the Fermi Large Area Telescope (LAT) to disentangle gamma-ray emission from the two components in the energy range from 10 GeV to 2 TeV, bridging the gap between previous measurements at GeV and TeV energies. Methods. We determine the morphology of emission associated to Vela X separately at energies <100 and >100 GeV, and compare it to the morphology seen at other wavelengths. Then, we derive the spectral energy distribution of the two gamma-ray components over the full energy range. Results. The best overall fit to the LAT data is provided by the combination of the two components derived at energies <100 and >100 GeV. The first component has a soft spectrum, spectral index 2.19 ± 0.16−0.22+0.05 $2.19\pm0.16^{+0.05}_{-0.22}$ 2.19±0.16−0.22+0.05 , and extends over a region of radius 1.°36±0.°04, consistent with the size of the radio nebula. The second component has a harder spectrum, spectral index 0.9 ± 0.3−0.1+0.3 $0.9\pm0.3^{+0.3}_{-0.1}$ 0.9±0.3−0.1+0.3 , and is concentrated over an area of radius 0.°63±0.°03, coincident with the X-ray cocoon that had already been established as accounting for the bulk of the emission at TeV energies. Conclusions. The spectrum measured for the low-energy component corroborates previous evidence for a roll-over of the electron spectrum in the extended radio nebula at energies of a few tens of GeV possibly due to diffusive escape. The high-energy component has a very hard spectrum: if the emission is produced by electrons with a power-law spectrum, the electrons must be uncooled, and there is a hint that their spectrum may be harder than predictions by standard models of Fermi acceleration at relativistic shocks.