Let $X$ be a separable Banach space, $M \subset X$ a linear subspace. Must $M$ be a Borel set in $X$?

I believe the answer is "no," since I have seen authors who are careful to talk about "Borel subspaces". But I have not been able to find a counterexample.

If the answer is indeed "no", does every infinite-dimensional separable Banach space contain a non-Borel dense linear subspace?