Living organisms exhibit persistent autonomy through internally generated goals and self-sustaining behavioral organization, yet current embodied agents remain driven by externally scripted objectives. This dependence on predefined task specifications limits their capacity for long-term deployment in dynamic, unstructured environments where continuous human intervention is impractical. We propose that personality traits provide an intrinsic organizational principle for achieving persistent autonomy. Analogous to genotypic biases shaping biological behavioral tendencies, personalities enable agents to autonomously generate goals and sustain behavioral evolution without external supervision. To realize this, we develop PEPA, a three-layer cognitive architecture that operates through three interacting systems: Sys3 autonomously synthesizes personality-aligned goals and refines them via episodic memory and daily self-reflection; Sys2 performs deliberative reasoning to translate goals into executable action plans; Sys1 grounds the agent in sensorimotor interaction, executing actions and recording experiences. We validate the framework through real-world deployment on a quadruped robot in a multi-floor office building. Operating without reliance on fixed task specifications, the robot autonomously arbitrates between user requests and personality-driven motivations, navigating elevators and exploring environments accordingly. Quantitative analysis across five distinct personality prototypes demonstrates stable, trait-aligned behaviors. The results confirm that personality-driven cognitive architectures enable sustained autonomous operation characteristic of persistent embodied systems. Code and demo videos are available at https://sites.google.com/view/pepa-persistent/.