This paper concerns the drainage of a thin liquid lamella into a Plateau border. Many models for draining soap films assume that their gas-liquid interfaces are effectively immobile. Such models predict a phenomenon known as “marginal pinching”, in which the film tends to pinch in the margin between the lamella and the Plateau border. We analyse the opposite extreme, in which the gas-liquid interfaces are assumed to be stress-free. We apply a nonlocal coordinate transformation that transforms the nonlinear governing equations into the linear heat equation. We thus obtain a travelling-wave solution and show that it is globally attractive. We therefore find that no marginal pinching occurs in this case: the film always approaches a monotonic profile at large times. This behaviour reflects a marked qualitative difference between a film with immobile interfaces and one with perfectly mobile interfaces. In the former case, suction of fluid into the Plateau border is localised, while in the latter it is instantaneously transmitted throughout the film.