The body of Marie McCracken was discovered inside a home in Old Coach Road, East Kilbride.
Emergency services were called to the property, which is believed to belong to Ms McCracken's nearby neighbour, at about 10.45pm on Wednesday, following reports that a woman was injured.
However, Ms McCracken was pronounced dead at the scene.
A post-mortem was being carried out to find out exactly how she died.
A police spokeswoman said: "We are following a definite line of inquiry."
Ms McCracken was the senior service assistant for four years at the WRVS shop at Hairmyres Hospital in the town.
Residents spoke of their shock at her death.
One woman, who did not want to be named, said: ""I was woken at about 2am, looked out and saw the street swarming with police.
"It's terrifying to think something like this can happen on your doorstep."
Uniformed officers stood guard at the house and a cordon was in place while investigations were carried out.
Detective Inspector David Tassie, from Hamilton CID, said: "From extensive inquiries that have taken place so far, I would like to reassure people I do not believe this was a random attack.
"A team of detectives has been assigned to this investigation and an incident room has been set up.
"Door to door inquiries are being carried out and forensic examinations are continuing to gather any information that can help our investigation.
"Extra uniform patrols are in the area."
Anyone who heard a disturbance or has information is asked to call police on 01355 564016 or Crimestoppers on 0800 555111.
In later editions of yesterday's paper we wrongly reported the victim's name after she had been identified locally. Police later identified the victim as Marie McCracken.