The force also said it is "engaged in a befitting retaliation" after the Rangers began unprovoked firing in the Pargwal area of Jammu since around 3 PM.
Earlier in the day, the force said, at least half-a-dozen mortar shells were fired from across the border and they exploded in the Dewra village of Sunderbani sector along the IB.
"The force retaliated strongly during cross-border firing and shot dead at least three Pakistan Rangers," a Border Security Force (BSF) spokesperson said.
Rao, who was drinking water, got hit just below his right ear.
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"Constable Rao was operated upon last night and is stable now," the spokesperson added.
Officials said such a hostile action between the two sides along the IB has been witnessed after a long time.
The two border guarding forces, on July 17, had held a Commandant-level flag meeting in Samba sector along the IB and "committed" themselves to maintaining peace.
Incidents of ceasefire violation by Pakistan have increased sharply this year.
Till August 1, there were 285 such violations by the Pakistan Army, while in 2016, the number was significantly less at 228 for the entire year, according to the Army figures.