In his first remarks since India and Pakistan both claimed to have shot down each other's fighter planes near the disputed border of Kashmir, the prime minister urged his countrymen to unite "as the enemy seeks to destabilise India".

"In the face of their objective, every Indian should stand as a wall, as a rock," Modi said.

"The entire country is one today and standing with our soldiers. The world is looking at our collective will and we have faith in our forces' capacity.

"India will live as one. India will work as one. India will grow as one. India will fight as one. India will win as one."

Modi did not mention Pakistan by name during the address to party workers.

But the rival neighbours have been urged by major world powers to exercise restraint after India and Pakistani fighter planes fought a brief skirmish Wednesday morning over the disputed Himalayan territory of Kashmir.

New Delhi said it lost one fighter jet and a pilot was missing in action, later confirming he was in Pakistani custody. India also downed a Pakistani jet, the foreign ministry said.

Pakistan claimed to have shot down two Indian jets and denied it lost any aircraft in the dogfight.

Islamabad said its incursion across the heavily militarised border was in response to Indian warplanes bombing Balakot, well inside its territory, on February 26.

That attack was the "jaw breaking" response Modi had promised after militants staged a suicide bombing in Indian Kashmir on February 14, killing at least 40 paramilitaries.

New Delhi blamed the Pakistan-based Jaish-e-Mohammad for the attack, and hit what it called a training camp for the militants in its aerial raid 12 days later.