I spent some time testing this morning...
The Facebook "Like" button works fine on all my websites, except for my vanilla 'admin' website that still uses the 2011/2012 themes. After setting the overflow property, the "Send" button worked well, but the "Like" button was still getting clipped. I tried a bunch of CSS tricks, but none worked. It's only after changing themes that it started working. So, in my experience, I've only been able to reproduce the problem on the 2011/2012 themes.
I've updated the FAQ with a little info on this issue. Hopefully the guys at Facebook will fix this issue, or some WordPress user will track down why it's only happening in the 2011/2012 themes (or I'm guessing themes that have used these as a starting point).
Let me know if you find a way around it.