Police spokesman Micky Rosenfeld said the shootings happened inside the bank, which was cordoned off with a large number of police. Israeli TV footage showed several ambulances parked outside. Four other people were wounded, one of them seriously.
Police said they were investigating the incident and backed off from initial claims that it had been a bungled bank robbery.
"We are not convinced entirely at this point that we are talking about a robbery or an attempted robbery but it could be we are talking about an incident where from the start, the murderer came in with an intention to shoot," national police chief Yohanan Danino told reporters.
The bank specialized in mortgages, and Rosenfeld said police were exploring whether the man had experienced financial difficulties or had any other personal dispute with bank workers. He said robbery also remained a possible motive. Rosenfeld said a gunfight broke out and the robber took one hostage, but as a standoff developed, he killed himself. The hostage, who was lightly wounded, walked out of the bank, Rosenfeld added.
He said police looked into the possibility that there had been a second suspect, but ruled that out. He also said police believed all of the dead had been killed by the gunman.