Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Sift together the flour, baking powder and salt into a medium bowl. Beat the butter, confectioners' sugar and lime zest in a large bowl with a mixer on medium-high speed until fluffy, about 2 minutes. Add the egg and vanilla and beat until combined. Reduce the mixer speed to medium low; add the flour mixture and beat until just combined. Transfer two-thirds of the dough to a separate bowl and set aside. Add the food coloring to the remaining one-third of the dough and beat with the mixer until combined.
Assemble a cookie press according to the manufacturer's instructions. Working in batches, fill the press with about 3 tablespoons plain dough, then 3 tablespoons green dough, then 3 more tablespoons plain dough. Press cookies about 1 inch apart onto 2 ungreased baking sheets.
Bake, switching the pans halfway through, until the cookies are set but not browned, 10 to 12 minutes. Let cool 3 minutes on the baking sheets, then transfer to racks to cool completely.