admissible estimators that are not Bayes

A question that popped up on X validated made me search a little while for point estimators that are both admissible (under a certain loss function) and not generalised Bayes (under the same loss function), before asking Larry Brown, Jim Berger, or Ed George. The answer came through Larry’s book on exponential families, with the two examples attached. (Following our 1989 collaboration with Roger Farrell at Cornell U, I knew about the existence of testing procedures that were both admissible and not Bayes.) The most surprising feature is that the associated loss function is strictly convex as I would have thought that a less convex loss would have helped to find such counter-examples.