  We study nearest-neighbour correlation functions for the ground state of the
supersymmetric XYZ spin chain with odd length and periodic boundary conditions.
Under a technical assumption related to the $Q$-operator of the corresponding
eight-vertex model, we show that they can be expressed exactly in terms of the
Painlev\'e VI tau functions $s_n$ and $\bar s_n$ introduced by Bazhanov and
Mangazeev. Furthermore, we give an interpretation of the correlation functions
in terms of the Painlev\'e VI Hamiltonian.
