There are various principles in Banach space theory that allow one to pass from a given sequence of vectors $(x_n)$, to a subsequence $(x_{n_k})$ with some desired property. I'm thinking here, in particular, of the Bessaga-Pelczynski selection principle which, given a Schauder basis $(e_n)$ for a Banach space $X$, allows the passage from a normalized sequence $(x_n)$ in $X$ which converges weakly to $0$, to a subsequence $(x_{n_k})$ which is congruent to a block basic sequence.

What I am wondering is if, given a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, and a sequence $(x_n)$ of unit vectors weakly converging to 0, can we find such a subsequence $(x_{n_k})$ as above so that the index set $\{n_k:k\in\mathbb{N}\}$ is in $\mathcal{U}$?

I would also be interested any related selection principles which can be done "along an ultrafilter", or reasons why one cannot hope for results like this.