The dealership will have agreed a part exchange value for your old car based on its condition and history, but it is their responsibility to check the vehicle and its history before providing this valuation.

If you had said your car had a full service history but you could not provide it for them that would be your responsibility, but if you have done this then we don't feel you have any further obligation to them. Also it is not clear what paperwork they are asking you to provide that would resolve this situation. If they continue to pursue this then we would suggest you ask them to put their claim in writing and contact Citizen's Advice.