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Perfectly normal non-metrizable non-Archimedean spaces are generalized Souslin lines

In this paper we prove the equivalence between the existence of perfectly normal, non-metrizable, non-archimedean spaces and the existence of “generalized Souslin lines”, i.e., linearly ordered spaces in which every collection of disjoint open intervals is σ-discrete, but which do not have a σ-discrete dense set.The key ingredient is the observation that every first countable linearly ordered space has a dense non-archimedean subspace.