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A Yemeni security official told Reuters arrests were continuing over the attack that killed 17 people, including six attackers, outside the heavily fortified embassy on Wednesday.
He declined to say how many people were held. Authorities said earlier 30 people suspected of belonging to Al Qaeda had been detained.
"I promise that we will chase them wherever they may be... and bring them to justice," Saleh said in a speech in the Red Sea port city of Hodeidah late on Thursday, which was published by state media.
"These terrorist actions are against everyone, not against the political system only, they are against development, stability and the Islamic religion," he said.
Susan el-Baneh, an 18-year-old American woman, and her Yemeni husband were killed while standing in line with relatives applying to visit the United States. The remaining dead were all Yemeni, apart from an Indian woman passer-by.
A group calling itself Islamic Jihad in Yemen, which is unrelated to the Palestinian group with a similar name, claimed responsibility.
The group said it belonged to Al Qaeda and vowed attacks on the British and Saudi embassies and on Yemeni officials unless Yemen freed several jailed members.
The British embassy in Sanaa will be closed on Saturday and Sunday, working days in Yemen, due to the current threats, a diplomat at the mission told Reuters.
The bombings were the biggest militant operation in Yemen since the attacks on the US warship Cole in 2000 and the French tanker Limburg in 2002.
The attackers were disguised in military uniforms and used cars similar to those driven by security forces in an attempt to get inside the embassy building.
The State Department has said the bombings bore "all the hallmarks" of an Al Qaeda attack but the United States had not yet concluded who was to blame.
The Yemeni government joined the US-led war against terrorism following the September 11 attacks on US cities in 2001.
It has jailed dozens of militants in connection with bombings of Western targets and clashes with authorities, but is still viewed in the West as a haven for Islamist militants.
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