Dorrie Thompson, 46, from Victoria, central London, suffered serious head injuries, including a fractured skull, when the man tried to steal her car as she loaded groceries into it.
The attack, at the Sainsbury supermarket in Nine Elms, south London, on Friday, was the second in 18 hours in the capital. Earlier, a woman had been threatened in the Waitrose car park at East Sheen.
Police believe that the attacker may have a history of mental illness.
Mrs Thompson, a voluntary church worker, has had two operations to remove blood clots from her brain and was described yesterday as being in a 'critical but stable' condition. Her husband, John, and three children were at her bedside.
The attack happened as she loaded food for a church barbecue into the boot of her BMW. Her assailant tried to steal the car but was deliberately blocked by another driver and fled.
A Sainsbury assistant, Sevim Isik, 55, was also hit when she tried to help. She said: 'He clubbed her like a hunter killing a baby seal. He said nothing, just lifted up the bat and hit the woman without showing any mercy.'
The man is described as black, with short Afro hair, in his mid- twenties, about 5ft 10in, medium build and wearing dark clothes. He may have some facial hair.
Police think the attack is linked to an incident in East Sheen on Thursday when a woman returning to her BMW in the Waitrose car park saw a man standing nearby with a baseball bat. He raised the bat and started to approach her, but walked away when two other shoppers appeared.
Detective Inspector Adrian Smale said: 'The nature of these incidents points to the possibility that we are dealing with a person with a history of mental illness. Everything points to the two incidents being linked. The descriptions of the suspect are similar.'
A spokesman for Sainsbury refused to comment on whether security would be increased.
He said: 'We never discuss security matters at all, but we are working closely with the police on the case.'Reuse content