The Palestine pound or Palestine lira, was the currency of the British Mandate of Palestine from 1 November 1927 to 14 May 1948, and of the State of Israel between 15 May 1948 and 23 June 1952, until it was replaced with the Israeli pound. The Palestine pound was also the currency of Transjordan until 1949 when it was replaced by the Jordanian dinar, and remained in usage in the West Bank of Jordan until 1950.
They had a currency.
There are multiple other instances of currencies that were issued by entities which are not countries. One example: Cryptocurrencies: Digital currencies like Bitcoin, Ethereum, and thousands of others are issued by decentralized networks or private entities rather than governments. They function independently of national monetary systems.
The confederate states had a currency also.
Palestine was a British territory, so it was essentially part of Britain.