Three robbers, one of whom "was certainly armed", entered a branch of the French Credit Agricole bank in Naples at around 11:30am, the source said.
They took hostage customers and employees, who were rescued some two hours later, he said.
"Thanks to the swift response... all the hostages were freed shortly after 1:30 pm, without serious injuries," Naples Prefect Michele di Bari said in a statement.
Police broke windows to get inside the bank, according to local media footage from the scene.
But the robbers escaped through a tunnel which led "into the sewers", the police source said.
"The thieves took the contents of dozens of safe deposit boxes", he said.
Asked how much the stolen goods were worth, he said: "Nobody but the clients knows what was in those boxes".
Some 40 police officers, including some with sniffer dogs, were searching the area for the robbers, while forensic police at the scene dusted for fingerprints, the source said.
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