The vehicle veered out of control after the brakes failed, the governor of Narino department, Luis Alfonso Escobar, told Blu Radio.
A video shared by the authorities showed rescue workers using ropes to try to reach the wounded at the bottom of a deep gorge.
Those with serious injuries were airlifted to hospital.
The troops were on their way to Narino, a department on the border with Ecuador that is plagued by cocaine trafficking, to take part in security operations.
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