In fact, the answer is "no".  Independence of the sample mean and variance characterizes the normal distribution.  This was shown by Eugene Lukacs in ["A Characterization of the Normal Distribution", The Annals of Mathematical Statistics, Vol. 13, No. 1 (Mar., 1942), pp. 91-93.][1]

I didn't know this, but Feller Vol II (1966, pg 86) says that R.C. Geary proved this, too.


  [1]: http://www.jstor.org/stable/2236166