Apparently the answer is no, not every connected Hausdorff Banach manifold is regular, not even when it is modeled on a separable Hilbert space.
I quote (verbatim) from J. Margalef-Roig, E. Outerelo-Dominguez, Differential Topology, North Holland Mathematics Studies 173, 1992, page 44f.
It is well known the result of General Topology that every
Hausdorff locally compact topological space satisfies the
Tychonoff axiom [M-O-P, V.2, pg. 231]. By this and the Riesz's
theorem every Hausdorff locally finite dimensional differentiable
manifold satisfies the Tychonoff axiom. This last affirmation is not
true for arbitrary Hausdorff differentiable manifolds. In
[M.O.1] there is an example of a Hausdorff connected
differentiable manifold $X$ of class $\infty$, such that $\partial X = \emptyset$, $X$ is not
regular and $X$ admits an atlas whose charts are modelled over an
infinite dimensional separable real Hilbert space.
They continue to add the regularity hypothesis in their results whenever it is needed.
The cited references are:
[M.O.P.] MARGALEF, J.-OUTERELO, E.-PINILLA, J.L.: Topologia,
I-V. Alhambra, Madrid 1975, 79, 79, 80 and 1982.
[M.O.1] MARGALEF, J.-OUTERELO, E.: Una variedad diferenciable
de dimension infinita, separada y no regular.
Rev. Mat. Hisp.-Am, IV, V.42, 1982, 51-55. (QuickView link).
Edit: As was pointed out by Benjamin Dickman in the comments, the example also appears in English in A. Kriegl, P.W. Michor, The convenient setting of global analysis, AMS (1997), 27.6 Non-regular manifold, page 266. The book is available as a pdf file on Kriegl's homepage.