Dense non-reflection for stationary collections of countable sets

We present several forcing posets for adding a non-reflecting stationary subset of Pω1 (λ), where λ ≥ ω2. We prove that PFA is consistent with dense non-reflection in Pω1 (λ), which means that every stationary subset of Pω1 (λ) contains a stationary subset which does not reflect to any set of size א1. If λ is singular with countable cofinality, then dense non-reflection in Pω1 (λ) follows from the existence of squares.