Is there any connection between "static correlation" and the Born-Oppenheimer approximation?
Credit to Rudi_Birnbaum for coming up in a comment to the question with what I think is the key here.

Static correlation is a difficult to grasp concept of which even a clear definition is beyond me. However, it seems clear that a full Configuration Interaction (FCI) calculation yields all correlation energy, thus, must correctly handle static correlation. From the discussion of the Born-Oppenheimer approximation (BO) by Jensen,$^{[1]}$ I gather that the BO approximation is essentially a modification of the Hamilton operator. This shortcoming cannot be healed by a FCI calculation that only considers the electronic problem. So, in this sense, the two concepts are unrelated.

(And now for the speculative part of my answer.)

However, I cannot speak as to whether they frequently show up together in chemical systems. Though I cannot find a reference for it, I recall that the beryllium atom requires a multi-reference treatment. Being a single atom that is rather heavy, the BO approximation could be acceptable for it. Thus, this would be a case where the two are not related.