Globular clusters (GCs) — compact and massive star clusters found ubiquitously around galaxies — are believed to be ancient relics (ages ≳ 10 Gyr) from the early formative phase of galaxies, although their physical origin remains widely debated. The most numerous GC populations are hosted by giant elliptical galaxies, where they can exhibit a broad dispersion in colour interpreted as a wide spread in metallicity. Here, we show that many thousands of similarly compact and massive super star clusters have formed at an approximately steady rate over, at least, the past ~1 Gyr around the nearby giant elliptical galaxy, NGC 1275, at the centre of the Perseus cluster. The number distribution of these young star clusters appears to exhibit a similar dependence in luminosity and mass as the even more numerous but older GCs around NGC 1275. In just a few Gyr, these super star clusters will evolve to become indistinguishable in broadband optical colours from the older GCs, and their spread in age add to the dispersion in colour of these GCs. The spatial distribution of the super star clusters resembles the filamentary network of multiphase gas in the cluster core, implying that they formed from molecular gas amassed from cooling of the hot intracluster gas. The sustained formation of super star clusters from cooled intracluster gas constitutes a previously unrecognised but prodigious source of GCs over cosmic timescales, and contributes to both their enormous numbers and broad colour dispersion in giant elliptical galaxies.