Photoionization of trapped atoms is a recent technique for creating ion beams with low transverse temperature. The temporal behavior of the current that can be extracted from such an ultracold ion source is measured when operating in the pulsed mode. A number of experimental parameters are varied to find the conditions under which the time-averaged current is maximized. A dynamic model of the source is developed that agrees quite well with the experimental observations. The radiation pressure exerted by the excitation laser beam is found to substantially increase the extracted current. For a source volume with a typical root-mean-square radius of 20 µm, a maximum peak current of 88 pA is observed, limited by the available ionization laser power of 46 mW. The optimum time-averaged current is 13 pA at a 36% duty cycle. Particle-tracking simulations show that stochastic heating strongly reduces the brightness of the ion beam at higher current for the experimental conditions.