Calcium-rich supernovae (SNe) represent a significant challenge for our understanding of the fates of stellar systems. They are less luminous than other SN types and they evolve more rapidly to reveal nebular spectra dominated by strong calcium lines with weak or absent signatures of other intermediate- and iron-group elements, which are seen in other SNe. Strikingly, their explosion sites also mark them out as distinct from other SN types. Their galactocentric offset distribution is strongly skewed to very large offsets (∼1/3 are offset >20 kpc), meaning they do not trace the stellar light of their hosts. Many of the suggestions to explain this extreme offset distribution have invoked the necessity for unusual formation sites such as globular clusters or dwarf satellite galaxies, which are therefore difficult to detect. Building on previous work attempting to detect host systems of nearby Ca-rich SNe, we here present Hubble Space Telescope imaging of five members of the class – three exhibiting large offsets and two coincident with the disc of their hosts. We find no underlying sources at the explosion sites of any of our sample. Combining with previous work, the lack of a host system now appears to be a ubiquitous feature amongst Ca-rich SNe. In this case the offset distribution is most readily explained as a signature of high-velocity progenitor systems that have travelled significant distances before exploding.