Ask for paper-thin slices of meat at the deli counter.
Cut apple into 1/4- to 1/2-inch slices (about 16). Toss together apple slices and lemon-lime soft drink to prevent browning; drain. Place 1 apple slice on top of 3 arugula leaves and 1 to 2 basil leaves.
Cut each prosciutto slice lengthwise into 4 strips. Wrap center of each apple bundle with 1 prosciutto strip. Arrange on a serving platter, and sprinkle with pepper.
Note: To make ahead, prepare recipe as directed. Cover bites with damp paper towels, and chill 30 minutes.