Let $X$ be a not necessarily separable (infinite-dimensional) Banach space and $D\subseteq X$ be dense linearly independent subset. Then does there exist a set of infinite-dimensional separable Banach subspaces $\{X\_i\}\_{i \in I}$ of $X$ with the property that:


* $D\cap X\_i$ is dense in $X\_i$,
* $\bigcup\_{i \in I} X\_i$ is dense in $X$?



Breaking up dense subset in non-separable space
For every countable subset $M\subseteq D$ set $X\_M=\overline{\text{span}(M)}$ (with the norm of $X$). These spaces seem to satisfy your requirements. Am I missing something? 


