The force required to squeeze a thin cylinder of paste between two approaching parallel circular plates depends upon the rheology of the paste and upon friction at the paste-plate interface. Pressure developed at the centre of the paste can move liquid relative to solids within the paste. We study the case in which the approach velocity of the plates is sufficiently slow that solid-liquid relative motion cannot be neglected during the timescale of the squeeze test. It is assumed that slip occurs at the paste-plate interface, and that the total stress within the paste may be predicted by a lubrication analysis. The deformation of the paste is close to a uniform straining motion. Gradients of liquid volume fraction across the narrow gap between the plates therefore develop slowly, and can be balanced by diffusion of liquid. The pressures and liquid volume fraction are therefore, to a first approximation, independent of the coordinate z normal to the plates. We consider two cases, both accessible to experiment. In the first, the plates are sufficiently large that the paste does not extend to their periphery. The radius of the region occupied by the constant volume of paste increases as the plates approach each other. In the second, the gap between plates of finite radius is always filled with paste which extrudes at the periphery of the plates as they approach each other. In both cases the liquid volume fraction is reduced in the central region of the paste near the axis, and is sometimes reduced to such a small value that the force required to push the plates together becomes exceedingly large. Numerical solution of the lubrication equations then requires very fine resolution near the axis, where the paste yield stress is high and pressure gradients are large. The contribution to the total force from the region near the axis becomes important and the lubrication approximation, which is poor in this region, becomes inadequate.