Police were called to Morrisons in Hilmore Way, Morecambe, at 10.55am on November 1.
A grey Audi A4 had been involved in a collision with a woman in the store car park.
The woman, 92, suffered a fractured hip and was taken by paramedics to Royal Lancaster Infirmary for treatment where she remains in a serious condition.
Police are appealing for information and urging any witnesses to come forward.
Sgt Marc Glass, of Lancashire Police, said: “An investigation is underway after a 92-year-old woman was seriously injured following a collision in Morecambe.
“Specially trained officers are in regular contact with the victim and her family.
“Unfortunately we have had a limited response to the original witness appeal issued on November 1.
“The supermarket was busy at the time of the collision and I believe that there may be witnesses with important information who have yet to contact the police.
“If you witnessed the incident, or saw either the woman or vehicle in the moments before the collision, please call us.”
Anyone with information can contact police on 101 quoting log reference 0532 of November 1.