We show that the existence of a continuous minimal sufficient statistic not equivalent to the order statistics, for $n \geqq 2$ independent observations, is not a sufficient condition for the family of densities, assumed to be Lipschitz, to be an exponential family. This result is intended to be compared with a theorem of Dynkin (p. 24 of ) which asserts that the existence of a sufficient statistic not equivalent to the order statistics implies that the family of densities is an exponential family, provided that the densities possess continuous derivatives.
J. L. Denny. "Note on a Theorem of Dynkin on the Dimension of Sufficient Statistics." Ann. Math. Statist. 40 (4) 1474 - 1476, August, 1969. https://doi.org/10.1214/aoms/1177697518