I asked this on MSE yesterday ( https://math.stackexchange.com/q/197873/39378 ) but no one has answered it yet. I hope it's not too soon to post it here.
Here are a few ways to formalize the question, so you can pick your favorite and answer it. Assume whatever large cardinals you like.
(1) Is it consistent with ZFC that there is an inaccessible cardinal $\delta$ and a nonempty finite set that is first-order definable without parameters over $(V_\delta,\in)$ but has no elements that are first-order definable without parameters over $(V_\delta,\in)$?
(2) Is there any model of ZFC that has a finite nonempty set, first-order definable without parameters over the model, with no element that is first-order definable without parameters over the model?
(3) Is it consistent with ZFC that there is an ordinal-definable finite nonempty set with no ordinal-definable member? (I am aware of the question A question about ordinal definable real numbers, but that question asks about sets of real numbers and I already know the answer to my question for sets of real numbers, or indeed for sets of subsets of any ordinal, because they are definably linearly ordered.)
(4) Any of the above formulations with ZFC replaced by ZF.