Did you try doing it again, making a new cylinder with everything reset to defaults? Maybe try chucking your Maya pref file. Looking back more closely at the screenshot, it's funky how that part is...

Ok, it was pretty simple.

Before you extrude, go to Polygons->Tool Options and select "Keep Faces Together"

This is because the faces aren't extruding as one piece, they're extruding separately. I'm actually having exactly the same problem right now with exactly the same thing. If I figure it out I'll...