The problem is that I need to use a button, switch is not allowed.
Bullshit. A plastic shirt button is NOT usable on an Arduino.
If you mean a push-button switch, the style of the switch matters not a whit.
Also the button need to speed up the fading proces with 100ms with every press.
So, look at the state change detection example to determine when the switch becomes pressed. When it does, increment or decrement some value. I can't imagine that adding 100 to some value used as the interval between changes is going to speed up anything. YMMV.