We define a separable Banach space $X$ and prove the existence of a $\bar \partial $-closed $C^\infty $-smooth $(0, 1)$-form $f$ on the unit ball $B$ of $X$, which is not $\bar \partial $-exact on any open subset. Further, we show that the sheaf cohomology groups $H^q(\Omega , \mathcal {0})=0$, $q\ge 1$, where $\mathcal {0}$ is the sheaf of germs of holomorphic functions on $X$, and $\Omega $ is any pseudoconvex domain in $X$, e.g., $\Omega =B$. As the Dolbeault group $H^{0,1}_{\;\bar \partial }(B)\not =0$, the Dolbeault isomorphism theorem does not generalize to arbitrary Banach spaces. Lastly, we construct a $C^\infty $-smooth integrable almost complex structure on $M=B\times \mathbb {C}$ such that no open subset of $M$ is biholomorphic to an open subset of a Banach space. Hence the Newlander–Nirenberg theorem does not generalize to arbitrary Banach manifolds.