Wheeler’s `spacetime-foam’ picture of quantum gravity (QG) suggests spacetime fuzziness (fluctuations leading to non-deterministic effects) at distances comparable to the Planck length, LPl ~ 1.62 × 10-33 cm, the inverse (in natural units) of the Planck energy, EPl ~ 1.22 × 1019 GeV. The resulting non-deterministic motion of photons on the Planck scale is expected to produce energy-dependent stochastic fluctuations in their speed. Such a stochastic deviation from the well-measured speed of light at low photon energies, c, should be contrasted with the possibility of an energy-dependent systematic, deterministic deviation. Such a systematic deviation, on which observations by the Fermi satellite set Planck-scale limits for linear energy dependence, is more easily searched for than stochastic deviations. Here, for the first time, we place Planck-scale limits on the more generic spacetime-foam prediction of energy-dependent fuzziness in the speed of photons. Using high-energy observations from the Fermi Large Area Telescope (LAT) of gamma-ray burst GRB090510, we test a model in which photon speeds are distributed normally around c with a standard deviation proportional to the photon energy. We constrain the model’s characteristic energy scale beyond the Planck scale at >2.8EPl(>1.6EPl), at 95% (99%) confidence. Our results set a benchmark constraint to be reckoned with by any QG model that features spacetime quantization.