An analysis of the motion and deformation of red blood cells between two parallel flat plates is presented. The motion is driven by an imposed pressure gradient in the surrounding fluid. Mammalian red cells are highly flexible, but deform at constant volume because the contents of the cell are incompressible, and at nearly constant surface area because the membrane strongly resists dilatation. Consequently, a minimum spacing between the plates exists, below which passage of intact cells is not possible. We consider spacings slightly larger than this minimum. The shape of the cell in this case is a disk with a rounded edge. The flow of the surrounding fluid is described using lubrication theory. Under the approximation that the distance between the plates is small compared with the cell diameter, cell shapes, pressure distributions, membrane stresses and cell velocities are deduced as functions of geometrical parameters. It is found that the narrow gaps between the cell and the plate are not uniform in width, and that as a result, membrane shear stresses are generated which increase in proportion to flow velocity. This contrasts with axisymmetric configurations, in which membrane shear stress remains bounded as cell velocity increases. The variation of cell velocity with spacing of the plates is similar to that previously demonstrated for rigid disk-shaped particles of corresponding dimensions.