Shelomo Ben-Yosef was the first Jew executed by the British in Palestine. Ben-Yosef was hung on June 29, 1938, at Acre prison for attempting to attack an Arab bus in reprisal for the murder of Jews at the hands of Arab terrorists.
Source: Bard, Mitchell G. and Moshe Schwartz. 1001 Facts Everyone Should Know About Israel. Rowman & Littlefield Publishers, 2005.