Adapting large language models (LLMs) to specialized financial reasoning typically requires expensive fine-tuning that produces model-locked expertise. Training-free alternatives have emerged, yet our experiments show that leading methods (GEPA and ACE) achieve only marginal gains on the FAMMA financial reasoning benchmark, exposing the limits of unstructured text optimization for complex, multi-step domain reasoning. We introduce Automated Skill Distillation and Adaptation (ASDA), a framework that automatically generates structured skill artifacts through iterative error-corrective learning without modifying model weights. A teacher model analyzes a student model's failures on financial reasoning tasks, clusters errors by subfield and error type, and synthesizes skill files containing reasoning procedures, code templates, and worked examples, which are dynamically injected during inference. Evaluated on FAMMA, ASDA achieves up to +17.33% improvement on arithmetic reasoning and +5.95% on non-arithmetic reasoning, substantially outperforming all training-free baselines. The resulting skill artifacts are human-readable, version-controlled, and compatible with the Agent Skills open standard, offering any organization with a labeled domain dataset a practical and auditable path to domain adaptation without weight access or retraining.