No, but if it travels fast enough it would disintegrate and you could argue the resulting plasma blast would be what actually damages the target.

Relevant XKCD (in video form).

Any matter going through you with that much mass is going to cause damage no matter how fast it goes. Billions of particles called neutrinos are moving through you right now as you read this but they are around 100,000,000,000,000x less massive than a hydrogen atom

Only in anime.

Define damage. Can it pass through the middle of organs? Sure, if it hits just right. But that's not so much a question of speed.

You'd need a pin needle shape to have a chance of piercing an organ without causing lasting damage, but it will probably break on impact

Not really applicable but think there could be a small chance it would quantum tunnel through the person but that's such miniscule chance.