There absolutely are. I work as a paramedic and among other things we have a contract with the state run secure mental health facility which has people who are civilly committed, but also people who have been adjudged not fit to stand trial.

It's also not entirely easy to get found not fit to stand trial - it's not just you and your lawyer and a friendly psychiatrist - the prosecution has the right to call its own mental health professionals and they are required to have access to the defendant, and more.