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We show that it is consistent from an inaccessible cardinal that classicalNamba forcing has the weak $\omega_1$-approximation property. In fact, this isthe case if $\aleph_1$-preserving forcings do not add cofinal branches to$\aleph_1$-sized trees. The exact statement we obtain is similar to Hamkins'Key Lemma. It follows as a corollary that Martin's Maximum implies that thereare stationarily many indestructibly weaky guessing models that are notinternally unbounded. This answers questions of Cox and Krueger. Our result onMartin's Maximum shortens the proof of Cox and Krueger's main result byremoving their use of higher Namba forcings, but we find another application oftheir ideas by answering a question of Adolf, Apter, and Koepke on preservationof successive cardinals by singularizing forcings.

Classical Namba forcing can have the weak countable approximation property