The first data release of the VELOcities of CEpheids project (VELOCE DR1, Anderson et al. 2024, A&A in press, arXiv: 2404.12280, doi: 10.1051/0004-6361/202348400) comprises 18,225 radial velocity measurements (RVs) of 258 bona-fide classical Cepheids as well as 1161 RVs of 164 additional targets, most of which were previously misclassified as Cepheids. The observations were collected mainly between 2010 and 2022 using two 1.2m telescopes: Euler (Coralie spectrograph) at ESO La Silla Observatory in Chile, and Mercator (Hermes spectrograph) at Roque de los Muchachos Observatory on La Palma, Canary Islands, Spain. Here, we publish the FITS files as described in appendix C of VELOCE paper I (Anderson et al. 2024). A total of 422 FITS files -- one per star -- are compressed together using tar and gzip as they would otherwise exceed the limit of 100 files in Zenodo. The FITS files contain all 19,386 individual RV measurements, as well as the per-epoch template fit residuals used to measure zero-point offsets and investigate long-term orbital motion.