No, not that
Many years ago, little ones, Henry Fowler made a suggestion that it would be a Good Thing to make a distinction—he loved distinctions—between that and which, reserving which for parenthetical clauses and using that for restrictive or limiting clauses.
Britons, who lightheartedly used which in both respects, continued to do so, but on These Shores the that/which distinction became a minor fetish among copy editors and the dear old, so frequently misguided, Associated Press Stylebook. Early in the course of writing this blog, I attempted a justification of the distinction, for which Geoffrey Pullum applied one of his gentle savagings.* Now I am as lighthearted as any Briton about which.
But I still wince when a construction like this lands on the desk: “The accident that took place near the Naval Academy remains under investigation.” There was one accident, Best Beloved, and its location is incidental to the sentence. There was not a spate of accidents compelling us to identify this one by its location.
And I see such nonrestrictive that clauses all the damn time.
So long as you are clear in context, you may indulge in whichery to your little hearts’ content. But stick to that only when you mean to single out one of two or more possibilities.
*No, I am not going to link back to it. Do I look crazy?
**Strike the question.