Mitchell County, NC has three ZIP codes:

28705, Bakersville, NC
28675, Penland, NC
28777, Spruce Pine, NC

Little Switzerland, part of which is in Mitchell County, has a ZIP code of 28749.  However, this ZIP technically belongs to McDowell County.

There was also a ZIP code (28740) for Relief, NC, which is in Mitchell County.  However that ZIP was  officially discontinued in 1974.