Personalized Federated Learning (PFL) enables collaboratively model training on decentralized, heterogeneous data while tailoring them to each client's unique distribution. However, existing PFL methods produce static models with a fixed tradeoff between accuracy and efficiency, limiting their applicability in environments where inference requirements vary with contexts and resource availability. Early-exit networks (EENs) offer adaptive inference by attaching intermediate classifiers. Yet integrating them into PFL is challenging due to client-wise heterogeneity and depth-wise interference arising from conflicting exit objectives. Prior studies fail to resolve both conflicts simultaneously, leading to suboptimal performance. In this paper, we propose CAFEDistill, a Conflict-Aware Federated Exit Distillation framework that jointly addresses these conflicts and extends PFL to early-exit networks. Through a progressive, depth-prioritized student coordination mechanism, CAFEDistill mitigates interference among shallow and deep exits while allowing effective personalized knowledge transfer across clients. Furthermore, it reduces communication overhead via a client-decoupled formulation. Extensive evaluations show that CAFEDistill outperforms the state-of-the-arts, achieving higher accuracy and reducing inference costs by 30.79%-46.86%.