Cut the ciabatta crosswise into 4 even pieces. Cut off a very thin layer from the top and bottom crusts. Halve each ciabatta horizontally; arrange 6 coppa slices on the bottom half of each sandwich, then top with 1 sliced cherry pepper and the arugula. Drizzle a little oil and vinegar over the arugula and season with salt and pepper. Close the sandwiches.
Heat 4 teaspoons of olive oil in each of 2 large cast-iron skillets. Add 2 of the sandwiches to each skillet and weight them down with a smaller cast-iron skillet or other heavy pan. Cook over moderately high heat until browned and crisp, 2 minutes per side. Cut the panini in half and serve hot.