The aim of the study was to determine the relative biological effectiveness (RBE) of a 60-MeV proton radiotherapy beam at the Institute of Nuclear Physics, Polish Academy of Sciences (IFJ PAN) in Kraków, the first one to operate in Poland. RBE was assessed at the surviving fractions (SFs) of 0.01, 0.1, and 0.37, for normal human fibroblasts from three cancer patients. The cells were irradiated near the Bragg peak of the pristine beam and at three depths within a 28.4-mm spread-out Bragg peak (SOBP). Reference radiation was provided by 6-MV X-rays. The mean RBE value at SF = 0.01 for fibroblasts irradiated near the Bragg peak of pristine beam ranged between 1.06 and 1.15. The mean RBE values at SF = 0.01 for these cells exposed at depths of 2, 15, and 27 mm of the SOBP ranged between 0.95–1.00, 0.97–1.02, and 1.05–1.11, respectively. A trend was observed for RBE values to increase with survival level and with depth in the SOBP: at SF = 0.37 and at the depth of 27 mm, RBE values attained their maximum (1.19–1.24). The RBE values estimated at SF = 0.01 using normal human fibroblasts for the 60-MeV proton radiotherapy beam at the IFJ PAN in Kraków are close to values of 1.0 and 1.1, used in clinical practice.