Brain lesion segmentation remains challenging due to small, low-contrast lesions, anisotropic sampling, and cross-slice discontinuities. We propose CenterMamba-SAM, an end-to-end framework that freezes a pretrained backbone and trains only lightweight adapters for efficient fine-tuning. At its core is the CenterMamba encoder, which employs a novel 3x3 corner-axis-center short-sequence scanning strategy to enable center-prioritized, axis-reinforced, and diagonally compensated information aggregation. This design enhances sensitivity to weak boundaries and tiny foci while maintaining sparse yet effective feature representation. A memory-driven structural prompt generator maintains a prototype bank across neighboring slices, enabling automatic synthesis of reliable prompts without user interaction, thereby improving inter-slice coherence. The memory-augmented multi-scale decoder integrates memory attention modules at multiple levels, combining deep supervision with progressive refinement to restore fine details while preserving global consistency. Extensive experiments on public benchmarks demonstrate that CenterMamba-SAM achieves state-of-the-art performance in brain lesion segmentation.