In a large bowl, cream butter and confectioners' sugar until blended. Beat in lime peel and extracts. In another bowl, mix flour and baking powder and salt; gradually beat into creamed mixture.
Stir in Cherries. Divide dough in half and shape each into long roll logs. Wrap in plastic wrap and refrigerate for a few hours or until firm.
Unwrap and cut dough into 1/4 inch slices. Place 2 inches apart on ungreased cookie sheet. Bake at 350 for 9 minutes or until edges are slightly golden. Remove pans and let cool slightly before removing.