Detection of sodium-salt-rich ice grains emitted from the plume of the Saturnian moon Enceladus suggests that the grains formed as frozen droplets from a liquid water reservoir that is, or has been, in contact with rock. Gravitational field measurements suggest a regional south polar subsurface ocean of about 10 kilometres thickness located beneath an ice crust 30 to 40 kilometres thick. These findings imply rock-water interactions in regions surrounding the core of Enceladus. The resulting chemical `footprints' are expected to be preserved in the liquid and subsequently transported upwards to the near-surface plume sources, where they eventually would be ejected and could be measured by a spacecraft. Here we report an analysis of silicon-rich, nanometre-sized dust particles (so-called stream particles) that stand out from the water-ice-dominated objects characteristic of Saturn. We interpret these grains as nanometre-sized SiO2 (silica) particles, initially embedded in icy grains emitted from Enceladus' subsurface waters and released by sputter erosion in Saturn's E ring. The composition and the limited size range (2 to 8 nanometres in radius) of stream particles indicate ongoing high-temperature (>90 °C) hydrothermal reactions associated with global-scale geothermal activity that quickly transports hydrothermal products from the ocean floor at a depth of at least 40 kilometres up to the plume of Enceladus.