  The N-extended supersymmetric self-dual Poincar\'e supergravity equations
provide a natural local description of supermanifolds possessing hyperk\"ahler
structure. These equations admit an economical formulation in chiral
superspace. A reformulation in harmonic superspace encodes self-dual
supervielbeins and superconnections in a graded skew-symmetric supermatrix
superfield prepotential satisfying generalised Cauchy-Riemann conditions. A
recipe is presented for extracting explicit self-dual supervielbeins and
superconnections from such `analytic' prepotentials. We demonstrate the method
by explicitly decoding a simple example of superfield prepotential, analogous
to that corresponding to the Taub-NUT solution. The superspace we thus
construct is an interesting $N=2$ supersymmetric deformation of flat space,
having flat `body' and constant curvature `soul'.
