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We prove that zero-sum Dynkin games in continuous time with partial andasymmetric information admit a value in randomised stopping times when thestopping payoffs of the players are general \cadlag measurable processes. As aby-product of our method of proof we also obtain existence of optimalstrategies for both players. The main novelties are that we do not assume aMarkovian nature of the game nor a particular structure of the informationavailable to the players. This allows us to go beyond the variational methods(based on PDEs) developed in the literature on Dynkin games in continuous timewith partial/asymmetric information. Instead, we focus on a probabilistic andfunctional analytic approach based on the general theory of stochasticprocesses and Sion's min-max theorem (M. Sion, Pacific J. Math., 8, 1958, pp.171-176). Our framework encompasses examples found in the literature oncontinuous time Dynkin games with asymmetric information and we providecounterexamples to show that our assumptions cannot be further relaxed.

On the value of non-Markovian Dynkin games with partial and asymmetric information