We define and study the independent natural extension of two local uncertainty models for the general case of infinite spaces, using the frameworks of sets of desirable gambles and conditional lower previsions. In contrast to Miranda and Zaffalon (2015), we adopt Williams-coherence instead of Walley-coherence. We show that our notion of independent natural extension always exists - whereas theirs does not - and that it satisfies various convenient properties, including factorisation and external additivity. The strength of these properties depends on the specific type of epistemic independence that is adopted. In particular, epistemic event-independence is shown to outperform epistemic atom-independence. Finally, the cases of lower expectations, expectations, lower probabilities and probabilities are obtained as special instances of our general definition. By applying our results to these instances, we demonstrate that epistemic independence is indeed epistemic, and that it includes the conventional notion of independence as a special case.