Residents of the complex said they could hear soldiers pleading with the insurgents to free women and children, and the children screaming, during the fighting.
The attack on the sprawling complex, which also houses a joint NATO-Afghan base, is the second major Taliban assault in the space of 24 hours in the city recognised as the birthplace of the Taliban.
The government claimed today that an unknown number of assailants had been killed but local residents, who were told to hunker down in their homes, were still reporting gunfire and explosions.
Dawood Shah Wafadar, a military commander in Kandahar, gave a higher death toll of 18.
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"The fighting started around 6:00 pm (yesterday) and intensified over the night," 30-year-old university student Izatullah, who lives inside the complex, told AFP.
"Soldiers were calling on Taliban attackers to let women and children go but attackers declined. We could hear children screaming during the fighting."
Some passengers were also trapped inside the civilian terminal, far from the fighting in the sprawling complex, when their commercial flight to India was suspended, Kandahar airport director Ahmadullah Faizi told AFP.
Ghani's willingness to visit longtime nemesis Pakistan has signalled a renewed push to jumpstart peace talks with the Taliban, despite a spike in cross-border tensions.
The Taliban claimed responsibility for the attack, with spokesman Zabihullah Mujahid saying on Twitter that "150 Afghan and foreign soldiers" had been killed in the fierce fighting.
The insurgents are regularly known to exaggerate battlefield claims.
The raid also comes after days of fevered speculation about the fate of Taliban leader Mullah Akhtar Mansour following reports that he was critically wounded in a firefight with his own commanders in Pakistan.