Police were called to reports of an altercation at Morrisons on Norwich Road, Wymondham, Norfolk, at 20:00 BST on Monday.
A local man in his 20s was treated by paramedics at the scene before being airlifted to the Norfolk and Norwich University Hospital.
Four men and a 19-year-old woman have been arrested.
Norfolk Police said the victim later died in hospital. His family are being supported by specially trained officers and a post-mortem examination will be held later.
Three of the men, aged 19, 41 and 42, and the woman were arrested on Monday night in connection with the incident, while a fifth person, a 43-year-old man, was arrested in the early hours of Tuesday. All remain in custody.
Det Ch Insp Marie James, from the Norfolk and Suffolk Major Investigation Team, said: "While we're in the early stages, this is a fast paced enquiry and a number of people are currently being questioned.
"We believe the people involved in this incident are known to one another."
He said to help address local concerns over this  "shocking" and "rare incident" there would be increased police patrols.
Anyone who witnessed the incident is urged to call Norfolk Police.
A murder inquiry has been launched after a man assaulted at a supermarket died in hospital.