For a pair of Banach spaces $X,Y$ and an ultrafilter $U$ it is easy to find an isomorphism between $(X\oplus Y)_U$ and $X_U\oplus Y_U$. Is this preserved under infinite sums, that is,
Let $X$ be an infinite-dimensional Banach space and let $U$ be an ultrafilter over $\mathbb N$. Do we have
$$\ell_\infty(X_U) \approx [\ell_\infty(X)]_U?$$
$\ell_\infty(X)$ denotes the $\ell_\infty$ sum of countably many copies of $X$.