The ELIMAIA-ELIMED beamline, powered by the L3 HAPLS petawatt laser, enables the irradiation of biological samples with intermediate-energy laser-driven protons (LDP) in a multi-shot regime. In the pilot radiobiological experiment, protons with a mean energy of similar to 24 MeV and doses up to similar to 14 mGy per shot, with similar to 4 ns bunch duration, were used to irradiate AG01522 normal human skin fibroblasts. The shortest irradiation time achieved was down to similar to 17 min/Gy, while the mean and peak dose rates reached similar to 1 x 10-3 and 3.5 x 106 Gy/s, respectively. The cells were exposed to doses ranging from similar to 0.4 to 1.5 Gy and analyzed for DNA damage, with double-strand breaks visualized as 53BP1 foci. Despite the differences in shot exposures between the multi-shot LDP and the previous experiments (at other facility) with single-shot LDP, similar DNA damage responses were observed. Results with conventionally accelerated protons align closely with the corresponding single-shot LDP samples. These experimental results were achieved as part of the flagship experiment FLAIM (within the IMPULSE EU-funded project) and serve as an initial demonstration of the ELIMAIA-ELIMED platform's potential for advanced radiobiological research, creating new opportunities for such studies utilizing laser-driven ion sources