No,He is not a white walker. Benjen was attacked by white walkers. They shoved an ice sword into his gut and left him for dead. He was discovered by the children of the forest who saved him and stopped him from becoming a wight by shoving a piece of Dragon glass into his chest.

Was Benjen stark a Wight?

You guessed it, Benjen is a White Walker...

They shoved a piece of dragonglass in his chest, which stopped him from falling to the control of the unfriendly White Walkers. That's why he was able to maintain loyalty to his family and help Bran and Meera.

Is Benjen stark Coldhands?

But Benjen isn't Coldhands. Although the theory seems logically consistent, it's basically the only Winds of Winter theory that has been definitively debunked. And that's because George R. R. Martin himself said so in a private note to his own editor, where he would have no reason to be hand-wavey and mysterious.