The 2006 competition is open. The BA PDFs are three-year research fellowships in the humanities and social sciences, “designed as early career opportunities for recently postdoctoral scholars to develop experience of research and teaching in the university environment”.
You must either have recently completed your PhD (viva on or after 1 July 2004) or do so by 30 June 2006, and have ‘a prior connection with the UK academic community’ (primarily through British citizenship and/or a PhD from a UK university), but must not have held ‘an established teaching post’ in a higher education institution. And of course, you need to have a well-thought-out research project proposal. Remember: it’s all about grantsmanship.
The competition is stiff, but if this is what you really want to do with your life, it’s worth a try.