If necessary, warm the wrap in a 350°F oven for 2 minutes to soften before filling. Lay the wrap on a work surface and spread the mayonnaise all over. Sprinkle the spinach leaves, cranberries, and carrots evenly on top. Arrange the Swiss cheese and turkey in even layers over the vegetable layer. Fold in the side edges and then roll up snugly from the bottom. Cut crosswise into 4 even pieces and wrap tightly in plastic.