BENGHAZI, Libya—The Libyan government arrested four people Thursday in connection with the deadly attack on the American consulate Tuesday night as Libyan and U.S. officials mounted a manhunt for others believed to be involved.

Protests spread across the region, breaking out in Yemen and Iran and once again in Cairo, where Egyptian police in riot gear beat back crowds of young men in a street filled with tear gas outside the U.S. Embassy.

In...