The Hurewicz dichotomy for generalized Baire spaces

By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space X is covered by a K_\sigma subset of X if and only if it does not contain a closed-in-X subset homeomorphic to the Baire space {}^\omega \omega. We consider the analogous statement (which we call the Hurewicz dichotomy) for \Sigma^1_1 subsets of the generalized Baire space {}^\kappa \kappa for a given uncountable cardinal \kappa with \kappa = \kappa^{<\kappa}. We show that the statement that this dichotomy holds at all uncountable regular cardinals is consistent with the axioms of ZFC together with GCH and large cardinal axioms. In contrast, we show that the dichotomy fails at all uncountable regular cardinals after we add a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the \kappa-perfect set property, the \kappa-Miller measurability, and the \kappa-Sacks measurability.