Extending Uniformly Continuous Pseudo-Ultrametrics and Uniform Retracts

It is first proved that any uniformly continuous pseudo-ultrametric on a subspace of a non-Archimedean uniform space X has a uniformly continuous extension to X (which preserves total boundedness or separability). Then it is proved that every complete subspace of an ultrametrizable space X is a uniform retract of X. This has consequences concerning the extension of uniformly continuous functions.