By the principle of local reflexivity, the second dual $X^{**}$ of a Banach space $X$ is complemented in some ultrapower $X^U$ of $X$. Even when $X$ is separable, the index set of $U$ cannot be expected to be countable. Indeed, for $X=\ell_1$ and an ultrafilter $U$ over a countable set, $X^U$ has cardinality continuum, yet $X^{**}$ has cardinality $2^{\mathfrak{c}}$.
Let us consider countably incomplete, non-principal ultrafilters $U$.
Suppose that for some such ultrafilter $U$, the canonical embedding of $X$ into $X^U$ has complemented range. Can we conclude that $X$ is complemented in $X^{**}$?
