We show that multi-window Gabor frames with windows in the Wiener algebra
$W(L^{\infty}, \ell^{1})$ are Banach frames for all Wiener amalgam spaces. As a
byproduct of our results we positively answer an open question that was posed
by [Krishtal and Okoudjou, Invertibility of the Gabor frame operator on the
Wiener amalgam space, J. Approx. Theory, 153(2), 2008] and concerns the
continuity of the canonical dual of a Gabor frame with a continuous generator
in the Wiener algebra. The proofs are based on a recent version of Wiener's
$1/f$ lemma.