According to this page
http://www.direct.gov.uk/en/Motoring/Dr ... DG_4022091 you can apply with any full passport; usually asylum-seekers have had to surrender their documents when applying for asylum, and don't expect the BIA to be sympathetic or efficient if you ask for its return. Asylum papers, ARC etc are not sufficient proof of ID - they only summarise what the applicant has claimed to be their details.
If you're receiving NASS support, it is possible there will be questions about how you can afford to drive and whether you are or are intending to work without permission.
I know someone who sent his ARC - stating no permission to work - along with payslips as additional evidence. Someone at DVLA was awake enough and bothered enough to contact the Home Office. His NASS support was terminated, but surprisingly he wasn't done for illegal working and defrauding NASS. He went on a year later to get papers under case resolution (he had wife and child here, resident about four years).
The Refused are coming day-by-day nearer to freedom.
