I'm going to leave my friend in the rain when I have an umbrella
Why wouldn't you share the umbrella with your friend?
Because I don't want to risk getting wet!
I think you should be fine sharing it.  It's wrong to let a friend suffer like that on the slight chance you might be slightly uncomfortable.  Learn to be a little bit more caring!
What if my friend wants to be wet and ends up making fun of me for being a pussy?
While that situation should arise, I think it's best to still be polite and extend your umbrella do them. Don't you want them to stay warm and not get sick? Why not ask them first if they want to get wet, so you know whether or not to share it.