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ABSTRACT We present Hubble Space Telescope ACS/WFC and WFC3/UVIS imaging for a sample of 50 low-surface brightness (LSB) galaxies in the $$\sim 10^{15}$$ M$$_{\odot }$$ Perseus cluster, which were originally identified in ground-based imaging. We measure the structural properties of these galaxies and estimate the total number of globular clusters (GCs) they host. Around half of our sample galaxies meet the strict definition of an ultra-diffuse galaxy (UDG), while the others are UDG-like but are either somewhat more compact or slightly brighter. A small number of galaxies reveal systems with many tens of GCs, rivalling some of the richest GC systems known around UDGs in the Coma cluster. We find the sizes of rich GC systems, in terms of their half-number radii, extending to $$\sim$$1.2 times the half-light radii of their host galaxy on average. The mean colours of the GC systems are the same, within the uncertainties, as those of their host galaxy stars. This suggests that GCs and galaxy field stars may have formed at the same epoch from the same enriched gas. It may also indicate a significant contribution from disrupted GCs to the stellar component of the host galaxy as might be expected in the ‘failed galaxy’ formation scenario for UDGs.