Whisk lime juice, chili paste, and sugar together in a small bowl. Add oil in slow stream while still whisking.
Combine napa cabbage, carrots, bell pepper, cucumber, cilantro, and peanuts in a large bowl. Season with salt and pepper. Pour dressing over vegetables and toss with tongs to coat. Chill salad for at least 1 hour. Stir and adjust seasonings before serving.