We report on the first systematic characterization of a tuneable laser-driven electron source capable of delivering Gy-scale doses in a duration of 10-20 ps in a single irradiation, thus reaching unprecedented dose rates in the range of 10^{10}-10^{12} Gy/s. Detailed characterization of the source indicates, in agreement with Monte Carlo simulations, dose delivery over cm-scale areas with a high degree of spatial uniformity. The results reported here confirm that a laser-driven source of this kind can be used for systematic studies of the response of biological cells to picosecond-scale radiation at ultrahigh dose rates.