We present the first experimental study of plasmoid formation in a magnetic reconnection layer undergoing rapid radiative cooling, a regime relevant to extreme astrophysical plasmas. Two exploding aluminum wire arrays, driven by the Z machine, generate a reconnection layer (S_{L}≈120) in which the cooling rate far exceeds the hydrodynamic transit rate (τ_{hydro}/τ_{cool}>100). The reconnection layer generates a transient burst of >1  keV x-ray emission, consistent with the formation and subsequent rapid cooling of the layer. Time-gated x-ray images show fast-moving (up to 50  km s^{-1}) hotspots in the layer, consistent with the presence of plasmoids in 3D resistive magnetohydrodynamic simulations. X-ray spectroscopy shows that these hotspots generate the majority of Al K-shell emission (around 1.6 keV) prior to the onset of cooling, and exhibit temperatures (170 eV) much greater than that of the plasma inflows and the rest of the reconnection layer, thus providing insight into the generation of high-energy radiation in radiatively cooled reconnection events.