Robots are increasingly envisioned as human companions, assisting with everyday tasks that often involve manipulating deformable objects. Recent advancements in robotic hardware and embodied AI algorithms have expanded the range of tasks robots can perform. However, current systems still struggle with handling thin, flat objects like paper and fabric due to limitations in motion planning and perception. This paper introduces PP-Tac, a robotic system designed specifically for handling paper-like objects. We developed a multi-fingered robotic hand equipped with high-resolution tactile sensors that provide omnidirectional feedback, enabling slippage detection and precise friction control with the material. Additionally, we created a grasp trajectory synthesis pipeline to generate a dataset of flat-object grasping motions and trained a diffusion policy for real-time control. This policy was then transferred to a real-world hand-arm platform for extensive evaluation. Our experiments, involving both everyday objects (e.g., plastic bags, paper, cloth) and more challenging materials (e.g., kraft paper handbags), achieved a success rate of 87.5%. By leveraging tactile feedback, our system also adapts to varying surfaces beneath the objects. These results demonstrate the robustness of our approach. We believe PP-Tac has significant potential for applications in household and industrial settings, such as organizing documents, packaging, and cleaning, where precise handling of flat objects is essential.