Pedagogical conversational agents (PCAs) such as chatbots and voice assistants can support learners in their studies. However, interactions with PCAs are often perceived as less motivating. Gamifying PCAs has been proposed as one approach to counteract this issue and increase learners' engagement. However, there is currently little prescriptive knowledge on how to design gamified PCAs. To address this, we conducted interviews with learners and reviewed relevant literature to derive 18 meta-requirements, five design principles, and 20 design features for gamified PCAs. We then applied our design knowledge to create a conceptual prototype, which we validated through an experiment with 76 participants. The experiment results demonstrate that our design knowledge can positively influence motivation and enjoyment in learning with PCAs.