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Highly unpredictable events appear to be abundant in life. However, when modeled rigorously, their existence in nature is far from evident. In fact, the world can be deterministic while at the same time the predictions of quantum mechanics are consistent with observations. Assuming that randomness does exist but only in a weak form, could highly random events be possible? This fundamental question was first raised by Colbeck and Renner~({\em Nature Physics}, 8:450--453, 2012). In this work, we answer this question positively, without the various restrictions assumed in the previous works. More precisely, our protocol uses quantum devices, a single weak randomness source quantified by a general notion of non-signaling min-entropy, tolerates a constant amount of device imperfection, and the security is against an all-powerful non-signaling adversary. Unlike the previous works proving non-signaling security, our result does not rely on any structural restrictions or independence assumptions. Thus it implies a stronger interpretation of the dichotomy statement articulated by Gallego et al.~({\em Nature Communications}, 4:2654, 2013): ``[e]ither our world is fully deterministic or there exist in nature events that are fully random.''