It is shown that even in very simple situations (like estimating the mean of a normal population) where a uniformly minimum variance unbiased estimator of the unknown population characteristic is known to exist, no best (even locally) unbiased estimator exists as soon as we alter slightly the definition of variance.
"A Note on the Theory of Unbiassed Estimation." Ann. Math. Statist. 26 (2) 345 - 348, June, 1955. https://doi.org/10.1214/aoms/1177728554