Speaking on condition of anonymity, an Arab Bank official said: "There were attacks against our buildings by civil servants angry at not receiving their pay and we had to close their doors."
Witnesses said there had been violent disputes between civil servants and a bank manager in one of the two branches.
Police officers were called after the disturbances, the witnesses said.
On Sunday, some cash-strapped Palestinian civil servants, who had not been paid since the end of February, began to receive money from their back salary cheques in their accounts at the Bank of Palestine.
About 40,000 of the Palestinian Authority's lowest-earning workers were eligible to receive the money promised by the Hamas-led government.
"I will have to pay some of my debts and the rest will go towards buying milk and basic food items. It's one good step but we need our full salaries"
a paramilitary policeman
Foreign donors and Israel have boycotted dealings with Hamas, which refuses to recognise Israel.
The boycott has meant Hamas has failed to meet payrolls for public employees since March.
Ala al-Maswabi, a paramilitary policeman, at the Cairo-Amman Bank in Gaza, said: "I will have to pay some of my debts and the rest will go towards buying milk and basic food items. It's one good step, but we need our full salaries."
All but one of about 20 banks in the West Bank and Gaza said they would use their own funds for the $13 million payout. They fear international sanctions if they deal with Hamas, viewed by the US as a terrorist organisation.
Hamas has promised advances to the rest of the government's 165,000 employees, but has set no date for payments.