The collision happened on the northbound carriageway near junction 44 for Aberford, between Leeds and Wetherby, at around 3pm on December 30.
A silver Volkswagen Polo struck a red Vauxhall Combo van.
West Yorkshire Police said: "A woman in her nineties, who was a passenger in the Combo van, was taken to hospital with serious injuries, which are said to be life-threatening due to her age. The drivers of both vehicles suffered minor injuries.
"Enquiries suggest the Polo had travelled the wrong way along the A64 from North Yorkshire before it joined the A1 carriageway, travelling southbound on the northbound carriageway.
"Anyone who saw the Polo on the A64, or who has dashcam footage that will assist is asked to contact the Roads Policing Unit on 101, quoting log 861 of 30 December."