A Silver forcing "below $2^n$" is defned e.g. in Definition 7.4.11 of [Tomek Bartoszyński and Haim Judah, Set Theory: on the structure of the real line, A. K. Peters, Wellesley, 1995.]. It is called *Infinitely equal forcing EE* there. In the same book, in Lemma 7.4.15 the authors show that

*EE preserves p-points*. However, the proof of this lemma seems to be not complete/correct: the choice of conditions $p^{n+1}$ is not clear, as needed extensions of $p^n$ for various $r_n^j$ may "contradict" each other. (This would not be a problem if conditions were $2^n$-splitting trees, i.e., if the forcing were more like the Sacks rather than Silver.)

Do you know another source/reference for the proof that EE preserves p-points? Or perhaps I am missing something and the proof in the book is actually complete?