  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
Hurewicz dichotomy) for $\Sigma^1_1$ subsets of the generalized Baire space
${}^\kappa \kappa$ for a given uncountable cardinal $\kappa$ with
$\kappa=\kappa^{<\kappa}$, and show how to force it to be true in a cardinal
and cofinality preserving extension of the ground model. Moreover, we show that
if the Generalized Continuum Hypothesis (GCH) holds, then there is a cardinal
preserving class-forcing extension in which the Hurewicz dichotomy for
$\Sigma^1_1$ subsets of ${}^\kappa \kappa$ holds at all uncountable regular
cardinals $\kappa$, while strongly unfoldable and supercompact cardinals are
preserved. On the other hand, in the constructible universe L the dichotomy for
$\Sigma^1_1$ sets fails at all uncountable regular cardinals, and the same
happens in any generic extension obtained by adding 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.
