The woman was found on the cycle path close to Ovangle Road at around 7.30am on Tuesday (April 5).
She was taken to hospital for treatment where police said she remained in a “serious condition” on Tuesday evening.
Police said it was unclear how the woman received her injuries but a man had been seen running from the area moments before.
Det Insp Andy Ellis, of Lancaster CID, said: “The victim, who was out walking her dog at around 7.30am, has suffered some serious injuries.
“We are taking this incident extremely seriously and are working hard to establish exactly what occurred.
“We are now appealing for information to help with our enquiries.”
The man who was spotted running away from the area was described as white and aged in his 20s.
He was wearing grey tracksuit bottoms and a light grey hooded top under a black coat.
Detectives would also like to speak to anyone who saw anything that seemed suspicious on the cycle path near Asda between 7.15am and 7.40am.
“We know the cycle path is busy with commuters and dog-walkers at this time of the morning so we believe there will have been people in the area who saw something potentially significant,” Det Insp Andy Ellis added.
“Whatever you know we would ask you to get in touch straight away.”
Anybody with information should contact the police by calling 101, quoting log number 212 of April 5.