Recent results have suggested that the statistics of bursts in the solar wind vary with solar cycle. Here, we show that this variation is basically absent if one considers extreme bursts. These are defined as threshold-exceeding events over the range of high thresholds for which their number decays as a power law. In particular, we find that the distribution of duration times and energies of extreme bursts in the solar wind ε parameter and similar observables are independent of the solar cycle and in this sense stationary, and show robust asymptotic power laws with exponents that are independent of the specific threshold. This is consistent with what has been observed for solar flares and, thus, provides evidence in favor of a link between solar flares and extreme bursts in the solar wind.