Authorities said at least nine other first responders were injured in the accident, some seriously.
"We must report, regrettably, that so far, five members of the federal police's firefighting unit have died, as have two members of the civil guard," Sergio Berni, Argentina's security secretary, told reporters.
Berni said rescuers were continuing to look for additional victims in the building's debris.
"Upon our arrival, we found a lot of bodies covered by rubble, some with multiple serious injuries, said Alberto Crescenti, head of the public medical service.
"We tried to get them out, but that was impossible," he said.
Officials said they do not know what caused the blaze in the 19th century era building, which was unoccupied at the time.
The wall fell as firefighters were using an electric saw to cut through a huge iron gate at the warehouse, which housed archival bank records.
The tragedy occurred in a neighbourhood of businesses and modest single family homes in southern Buenos Aires.
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