  We solve the classical problem of Plateau in every metric space which is
$1$-complemented in an ultra-completion of itself. This includes all proper
metric spaces as well as many locally non-compact metric spaces, in particular,
all dual Banach spaces, some non-dual Banach spaces such as $L^1$, all Hadamard
spaces, and many more. Our results generalize corresponding results of Lytchak
and the second author from the setting of proper metric spaces to that of
locally non-compact ones. We furthermore solve the Dirichlet problem in the
same class of spaces. The main new ingredient in our proofs is a suitable
generalization of the Rellich-Kondrachov compactness theorem, from which we
deduce a result about ultra-limits of sequences of Sobolev maps.
