Let $f:X\to Y$ be a morphism of Stein manifolds. Let $F$ be a coherent $O_X$-module, $\mathcal{A}=f_*O_X$. The question is: do we know that $f_*F$ is a coherent $\mathcal{A}$-module?
In fact, there is an algebraic analogue. Let $g:S\to T$ be an affine morphism of schemes, $\mathcal{B}:=g_*O_S$ then $g_*:Qch(O_S)\to Qch(\mathcal{B})$ (of quasi-coherent sheaves) is an equivalence of categories, see https://stacks.math.columbia.edu/tag/01SB. Hence, the restriction $g_*:Coh(O_S)\to Coh(\mathcal{B})$ (of coherent sheaves) is still an equivalence. In view of this, I wonder if it holds in the analytic case.