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Space-variant control of optical wavefronts is important for many applications in photonics, such as the generation of structured light beams, and is achieved with spatial light modulators. Commercial devices, at present, are based on liquid-crystal and digital micromirror technologies and are typically limited to kilohertz switching speeds. To realize significantly higher operating speeds, new technologies and approaches are necessary. Here we demonstrate two-dimensional control of free-space optical fields at a wavelength of 1,550 nm at a 1 GHz modulation speed using a programmable plasmonic phase modulator based on near-field interactions between surface plasmons and materials with an electrooptic response. High χ(2) and χ(3) dielectric thin films of either aluminium nitride or silicon-rich silicon nitride are used as an active modulation layer in a surface plasmon resonance configuration to realize programmable space-variant control of optical wavefronts in a 4 × 4 pixel array at high speed.