The linear stability of gravity-driven annular liquid film flow outside wires and inside tubes is analyzed when the rigid surface is replaced by a deformable (neo-Hookean) solid wall. On a rigid surface, an annular liquid diread becomes unstable due to a Rayleigh-type capillary instability even in the absence of flow. In the presence of flow, the annular liquid film becomes unstable due to a flow-driven, free-surface instability, which occurs over and above the curvature-induced capillary instability. In this paper, a low-wavenumber perturbation analysis is first used to elucidate the effect of wall deformability on the free-surface instability. It is shown that the free-surface instability is completely stabilized in the low-wavenumber limit when the wall is made sufficiently deformable. A numerical method is subsequently used to determine the stability of the system at arbitrary wavenumbers. Results from the numerical solution reveal that the prediction of instability suppression at low wavenumbers extends to finite wavenumbers as well. However, as the solid wall is made deformable even further, the free surface is destabilized at finite wavenumbers by the deformability; in addition, the liquid-solid interface could also become unstable when the solid deformability becomes high. It is demonstrated, however, that there is a sufficient range of shear modulus of the solid where the annular flow is stable at all wavenumbers. The results of this study have implications in wire-coating operations, as well as in biological settings such as closure of lung airways.