The steady-state extrusion coating of an extremely viscous power-law liquid from a slot and onto a moving substrate across a narrow gap is examined. A key feature of this configuration is a two-dimensional liquid sheet that bridges the gap and might be subjected to an ambient pressure drop across its wide faces. The local thickness of this sheet may be viewed as gradually thinning from the slot to the substrate, and simplified equations governing its shape are derived. Analytical solutions for the shape of the liquid sheet, and the flow within, are obtained in the limit where viscous and pressure forces dominate all others. It is found that the centerline shape of the liquid sheet is circular when a pressure drop is applied, despite the fact that surface tension forces, which typically give rise to circular shapes in narrow gaps, are neglected. Nevertheless, it is analytically shown that the stretching of the viscous-dominated liquid sheet gives rise to an effective tension that plays an analogous role to surface tension. The range of possible applied pressures is thus deduced via geometrical considerations of a circular shape constrained in the narrow gap, as in the analogous surface-tension-dominated analysis (Ruschak, 1976. Chemical Engineering Science 31, 1057); extrusion coating fails when a liquid sheet cannot be constructed successfully. (c) 2005 Elsevier Ltd. All rights reserved.