We present deep, wide-field European VLBI Network (EVN) 1.6 GHz observations of the Hubble Deep Field (HDF) region with a resolution of 0.025 arcseconds. Above the 210 microJy/beam (5sigma) detection level, the EVN clearly detects two radio sources in a field that encompasses the HDF and part of the Hubble Flanking Fields (HFF). The sources detected are: VLA J123644+621133 (a z=1.013, low-luminosity FR-I radio source located within the HDF itself) and VLA J123642+621331 (a dust enshrouded, optically faint, z=4.424 starburst system). A third radio source, J123646+621404, is detected at the 4sigma level. The VLBI detections of all three sources suggest that most of the radio emission of these particular sources (including the dusty starburst) is generated by an embedded AGN.