To add onto the correct answer by @Justin Ethier Benjen was neither wight nor White Walker but some other undead being saved from death/becoming a wight by the Children of the Forest.
BRAN: Uncle Benjen. The last letter Jon wrote me said you had been lost beyond the Wall.
BENJEN: I led a ranging party deep into the North to find White Walkers. They found us. A white walker stabbed me in the gut with a sword of ice. Left me there to die. To turn. The Children found me. Stopped the walker’s magic from taking hold.
BENJEN: The same way they made the Walkers in the first place. You saw it yourself.
BRAN: Dragonglass. A shard of dragonglass plunged into your heart.
Game of Thrones, S06 E06, "Blood of my Blood"
However, as of Season 7 Episode 6, "Beyond the Wall", he is presumed truly dead after sacrificing himself for Jon.
Go! Go now! Leave! Uncle Benjen! How? You ride for the pass.
Come with me.
It's not time.
Game of Thrones, S07 E06, "Beyond the Wall"
You can watch the scene here where he gets taken down by an overwhelming majority of wights: