GHz-peaked spectrum (GPS) radio sources are thought to be young objects which later evolve into FR-I and FR-II radio galaxies. We have used the Australia Telescope 20GHz (AT20G) survey catalogue to select a uniform sample of GPS sources with spectral peaks above 5GHz, which should represent the youngest members of this class. In this paper, we present e-VLBI observations of ten such objects which are associated with nearby (z<0.15) galaxies and so represent a new population of local, low--power GPS sources. Our e-VLBI observations were carried out at 4.8GHz with the Australia Telescope Long Baseline Array (LBA) using a real--time software correlator. All ten sources were detected, and were unresolved on scales of ~100mas, implying that they are typically less than 100pc in linear size.