  We prove that any suitable generalization of Laver forcing to the space $
\kappa^\kappa$, for uncountable regular $\kappa$, necessarily adds a Cohen
$\kappa$-real. We also study a dichotomy and an ideal naturally related to
generalized Laver forcing. Using this dichotomy, we prove the following
stronger result: if $ \kappa^{<\kappa}=\kappa$, then every
$<\kappa$-distributive tree forcing on $\kappa^\kappa$ adding a dominating
$\kappa$-real which is the image of the generic under a continuous function in
the ground model, adds a Cohen $\kappa$-real. This is a contribution to the
study of generalized Baire spaces and answers a question from arXiv:1611.08140
