Starting from a variational principle for perfect fluids, the authors develop a Hamiltonian formulation for perfect fluids coupled to gravity expressed in Ashtekar's spinorial variables. The constraint and evolution equations for the gravitational variables are at most quadratic in these variables, as in the vacuum case and in the coupling of gravity to other matter fields, while some of the matter evolution equations are in general non-polynomial. They specialize the formalism to barotropic fluids and spherically symmetric spacetimes, and, within this class, to Kantowski-Sachs spacetimes. They find explicitly the Kantowski-Sachs solutions corresponding to 'stiff matter', which they use as examples to look at the behaviour of the Ashtekar variables when the spatial metric becomes degenerate on one hypersurface. They find that in these solutions the coordinate time arising in the present treatment is singularly related to proper time, and the singularities are only reached at infinite values of the former. They obtain some simple necessary conditions that have to be satisfied if one wants to evolve data past singularities of this kind. None of the barotropic-fluid-filled Kantowski-Sachs spacetimes satisfy these conditions.