We present time-of-flight measurements of the longitudinal energy spread of pulsed ultracold ion beams, produced by near-threshold ionization of rubidium atoms captured in a magneto-optical atom trap. Well-defined pulsed beams have been produced with energies of only 1 eV and a root-mean-square energy spread as low as 0.02 eV, 2 orders of magnitude lower than the state-of-the-art gallium liquid-metal ion source. The low energy spread is important for focused ion beam technology because it enables milling and ion-beam-induced deposition at sub-nm length scales with many ionic species, both light and heavy. In addition, we show that the slowly moving, low-energy-spread ion bunches are ideal for studying intricate space charge effects in pulsed beams. As an example, we present a detailed study of the transition from space charge dominated dynamics to ballistic motion.