We report on sensitive dual-frequency (1.7 and 5 GHz) European very long baseline interferometry Network observations of the central region of nine Seyfert galaxies. These sources are among the faintest and least luminous members of a complete sample of nearby (d < 22 Mpc) low-luminosity active galactic nuclei. We detect radio emission on milliarcsecond scale in the nuclei of four galaxies, while for the other five sources we set an upper limit of <∼ 100 μJy. In three sources, namely NGC 3227, NGC 3982 and NGC 4138, radio emission is detected at both 1.7 and 5 GHz and it is resolved in two or more components. We describe the structural and spectral properties of these features; we find that in each of these three nuclei there is one component with high brightness temperature (typically TB > 107.5 K) and flat/intermediate spectral index (0.3 ≤ α ≤ 0.6, S(ν) ∼ ν−α), accompanied by secondary steep spectrum extended components. In these cases, non-thermal emission from jets or outflows is thus the most natural explanation. A faint feature is detected in NGC 4477 at 5 GHz; keeping in mind the modest significance of this detection (∼5 σ), we propose the hot corona as the origin of non-thermal emission, on the basis of the unrealistic magnetic field values required by synchrotron self-absorption. Finally, the five non-detected nuclei remain elusive and further observations on intermediate scales will be necessary to investigate their nature.