A man has been arrested on suspicion of attempted murder and robbery, in connection with the Dulwich Park stabbing incident which took place on Thursday.
The 26-year-old male suspect was arrested yesterday, but has yet to be formally charged.
The 31-year-old man who received a stab wound to his head remains in a critical condition, police have said.
Earlier reports suggested the victim had intervened when the suspect allegedly tried to rob a woman of her handbag.
Police have confirmed that the nearby Co-op supermarket in Forest Hill Road was also closed on Thursday, in connection with this investigation, but were unwilling to explain why.
Detective superintendent Helen Lyons, from Southwark Borough, said: “I would like to thank those in the community for coming forward with information as this has greatly assisted the investigation.
“Whilst we have made an arrest this evening I would urge those that have yet to come forward and speak to us to do so as the information you hold may be vital to progressing this investigation further.”
Police and Air Ambulance were called to the incident in Dulwich Park at 1:03pm.
It was also understood that either the victim, or the woman who was allegedly targeted for her handbag, was walking their dog at the time. The dog was reported to have run away, and was thought to be missing, but was found safe and well on the evening of the incident, according to reports on social media.
Any witnesses or anyone with any information is asked to call police on 101 or via Twitter @MetCC.
To give information anonymously contact Crimestoppers on 0800 555 111 or contact online at crimestoppers-uk.org.
