Can one take damage from getting shot in the hat?

In TF2, I usually wear smaller hats, especially as Sniper, to reduce my visual profile, so that I'm less likely to be seen.

But that leads me to another question: what if one is wearing a tall hat like Dr.'s Dapper Topper?

Obviously, a soldier's head only physically occupies the bottom half of Dr.'s Dapper Topper, so what happens when the top half is shot?

Does the soldier take damage from any shot anywhere in the hat?

Do non-set hats have any effect on where or how a player can be hit?

Solution 1:

No, hats are not part of the hitbox, in fact the hitbox for your head probably isn't quite where you think it is:

The head hitbox is actually quite a bit larger than the character's actual heads, take a peek at the hitbox renders: