Stringent observational constraints on the subgalactic matter power spectrum would allow one to distinguish between the concordance ΛCDM and the various alternative dark-matter models that predict significantly different properties of mass structure in galactic haloes. Galaxy–galaxy strong gravitational lensing provides a unique opportunity to probe the subgalactic mass structure in lens galaxies beyond the Local Group. Here, we demonstrate the first application of a novel methodology to observationally constrain the subgalactic matter power spectrum in the inner regions of massive elliptical lens galaxies on 1–10 kpc scales from the power spectrum of surface-brightness anomalies in highly magnified galaxy-scale Einstein rings and gravitational arcs. The pilot application of our approach to Hubble Space Telescope (HST/WFC3/F390W) observations of the SLACS lens system SDSS J0252+0039 allows us to place the following observational constraints (at the 99 per cent confidence level) on the dimensionless convergence power spectrum Δ2δκ and the standard deviation in the aperture mass σAM: Δ2δκ < 1 (σAM < 0.8 × 108 M⊙) on 0.5-kpc scale, Δ2δκ < 0.1 (σAM < 1 × 108 M⊙) on 1-kpc scale and Δ2δκ < 0.01 (σAM < 3 × 108 M⊙) on 3-kpc scale. These first upper-limit constraints still considerably exceed the estimated effect of CDM subhaloes. However, future analysis of a larger sample of galaxy–galaxy strong lens systems can substantially narrow down these limits and possibly rule out dark-matter models that predict a significantly higher level of density fluctuations on the critical subgalactic scales.