Deputy chief of government of Buenos Aires, Diego Santilli, said police had carried out 18 raids as part of the investigation into the death of Matthew Gibbard.
The 50-year-old and his stepson, Stefan Zone, 28, were targeted by a robbery gang outside their luxury hotel on Saturday morning.
Mr Gibbard was reportedly shot in the chest, while Mr Zone is recovering in hospital after being shot in the leg.
The politician denounced the killing, saying on Twitter: “What happened on Saturday was an atrocious event and these criminals are murderers.
"My sincerest condolences to the family of Matthew Charles Gibbard.”
Officials said the holidaymakers were shot as they tried to resist attempts to steal their belongings.
Footage shows a struggle outside the Faena Art Hotel before the suspects flee the scene on a motorcycle, leaving one man lying in the street.
Mr Gibbard was a director of Northamptonshire firm Tingdene, a company that makes holiday homes and lodges.
Fellow manufacturer Omar Park Homes posted a message on Twitter offering “heartfelt condolences” to Tingdene staff.
It said: “Our heartfelt condolences go out to the family, friends and all those at Tingdene for the loss of Matthew Gibbard at this incredibly sad time. We also wish Matthew’s stepson Stefan a fast recovery. Our thoughts are with you.”
Local media reports said police believe the motorbike robbers were supported by accomplices in a vehicle and are investigating whether the victims were followed to the hotel from the airport.
More than 111,000 British nationals visited Argentina in 2018, according to the Foreign Office, which said most visits are "trouble free".
But tourists are warned to be alert to street crime, including armed robberies, and advised to hand over cash and valuables without resistance.