First cut turns up nothing.
I haven't read all three pages of listings for "heart" in Bartlett's index yet, but "my", "be", "still" and "beating" have no entries. (Instinct tells me that "be still my heart" is probably the original, with "beating" added to later uses for good measure.)
Camp's Unfamiliar Quotations from 2000 B.C. to the Present also turns up nothing. Haven't hit The Home Book of Quotations yet.
It may be as Tom and Frank have said, that it's one of those common expressions that has no particular authorship.