In fact, the answer is "no". Independence of the sample mean and variance characterizes the normal distribution. This was shown by Eugene Lukacs in 1942, The Annals of Mathematical Statistics, Vol. 13, No. 1 (Mar., 1942), pp. 91-93.
I didn't know this, but Feller (1966, pg 86) says that R.C. Geary proved this, too.