BSF Deputy Inspector General (Malda Sector) Raj Singh Rathore said constable Basant Singh (35), posted at the 17th Mile troop centre of the force at Farakka, shot dead Head Constable Mool Chand in the early morning hours.
Three others, who sustained bullet injuries, were rushed to a private nursing home. Earlier, officials had said that four jawans were injured.
"Among the injured, two jawans are critical and under observation," a senior officer said.
Officials said the incident happened in the 20th battalion camp of the Border Security Force when the victims were asleep.
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BSF and local police have launched a massive hunt to nab Singh, a resident of Morena in Madhya Pradesh, who is absconding since the incident took place.
The injured personnel are constables Kamal Basa, S K Prabhakar and Surender.
"All the victims have sustained bullet injuries fired from the INSAS rifle of Singh. About 4-5 rounds are suspected to have been fired," the official said.