High-redshift radio-loud quasars are used to, among other things, test the predictions of cosmological models, set constraints on black hole growth in the early Universe and understand galaxy evolution. Prior to this paper, 20 extragalactic radio sources at redshifts above 4.5 have been imaged with very long baseline interferometry (VLBI). Here, we report on observations of an additional ten z > 4.5 sources at 1.7 and 5 GHz with the European VLBI Network, thereby increasing the number of imaged sources by 50 per cent. Combining our newly observed sources with those from the literature, we create a substantial sample of 30 z > 4.5 VLBI sources, allowing us to study the nature of these objects. Using spectral indices, variability and brightness temperatures, we conclude that of the 27 sources with sufficient information to classify, the radio emission from one source is from star formation, 13 are flat-spectrum radio quasars and 13 are steep-spectrum sources. We also argue that the steep-spectrum sources are off-axis (unbeamed) radio sources with rest-frame self-absorption peaks at or below GHz frequencies and that these sources can be classified as gigahertz peaked-spectrum and megahertz peaked-spectrum sources.