The central challenge in robotic manipulation of deformable objects lies in aligning high-level semantic instructions with physical interaction points under complex appearance and texture variations. Due to near-infinite degrees of freedom, complex dynamics, and heterogeneous patterns, existing vision-based affordance prediction methods often suffer from boundary overflow and fragmented functional regions. To address these issues, we propose TRACER, a Texture-Robust Affordance Chain-of-thought with dEformable-object Refinement framework, which establishes a cross-hierarchical mapping from hierarchical semantic reasoning to appearance-robust and physically consistent functional region refinement. Specifically, a Tree-structured Affordance Chain-of-Thought (TA-CoT) is formulated to decompose high-level task intentions into hierarchical sub-task semantics, providing consistent guidance across various execution stages. To ensure spatial integrity, a Spatial-Constrained Boundary Refinement (SCBR) mechanism is introduced to suppress prediction spillover, guiding the perceptual response to converge toward authentic interaction manifolds. Furthermore, an Interactive Convergence Refinement Flow (ICRF) is developed to aggregate discrete pixels corrupted by appearance noise, significantly enhancing the spatial continuity and physical plausibility of the identified functional regions. Extensive experiments conducted on the Fine-AGDDO15 dataset and a real-world robotic platform demonstrate that TRACER significantly improves affordance grounding precision across diverse textures and patterns inherent to deformable objects. More importantly, it enhances the success rate of long-horizon tasks, effectively bridging the gap between high-level semantic reasoning and low-level physical execution. The source code and dataset will be made publicly available at https://github.com/Dikay1/TRACER.