Does immunity to piercing damage make you immune to the pull effect of the spell Thorn Whip?


If a character has gained immunity to all piercing damage, would that character also be immune to the pull effect of Thorn Whip?

For reference:

Thorn Whip
Transmutation cantrip

Casting Time: 1 action
Range: 30 feet
Components: V, S, M (the stem of a plant with thorns)
Duration: Instantaneous

You create a long, vine-like whip covered in thorns that lashes out at your command toward a creature in range. Make a melee spell attack against the target. If the attack hits, the creature takes 1d6 piercing damage, and if the creature is Large or smaller, you pull the creature up to 10 feet closer to you. This spell’s damage increases by 1d6 when you reach 5th level (2d6), 11th level (3d6), and 17th level (4d6).