Let X be a Banach space, and let Y be a proper non-meager subspace of X. If Y is not dense in X, then it is easy to see that the closure of Y has empty interior, contradicting Y being non-meager. So Y must be dense. If Y has the Baire property, then it follows from Pettis Lemma that Y is open and hence closed, contradicting Y being proper. Thus, Y must be dense and not have the Baire property. My question is: can such a Y exist? Any such Y must be difficult to construct since all Borel sets and even all continuous images of separable complete metric spaces have the Baire property.