Hazel McLachlan was last seen leaving the hospital around 5pm on Monday, and has not been heard from since.
The 54-year-old, who is diabetic, is described as white, 5ft 6in, heavy build, with short grey hair, and a ruddy complexion. She usually wears chains around her neck and rings on every finger. When last seen she was wearing jeans and a T-shirt.
A police spokesman said: “Hazel depends on insulin to manage her diabetes, and we have concerns over her current health and well-being.
“There is a possibility she may have been back to her home address in the Dalry area, however she is also thought to have links to Selkirk and may have travelled there.
“Anyone who has any information regarding her current whereabouts or who has seen or heard from her should get in touch with police urgently.”