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Spiking neural networks (SNNs) exploit event-driven and addition-only computation to substantially improve efficiency for intelligent computation. A key temporal property of SNNs, elastic inference, allows outputs to emerge progressively, enabling responses to salient inputs much earlier than full evaluation. However, existing SNN-specific accelerators cannot capitalize on this property. Layer-by-layer designs emit outputs only after all layers are complete, while time-step-by-time-step designs rely on coarse-grained, layer-wise pipelines that require synchronizing all spines/tokens within a layer. This barrier prevents results from being forwarded immediately, delaying the earliest possible response and forfeiting the benefits of elastic inference. To address these challenges, we propose ELSA, a near-SRAM dataflow architecture that realizes true elastic inference through a fine-grained spine/token-wise pipeline and hardware optimizations tailored to SNNs. ELSA forwards each spine/token immediately upon production, forming a continuous streaming pipeline that substantially reduces the latency to the first response. To enhance this lightweight execution, ELSA introduces a bundled address event representation protocol to lower communication traffic of network-on-chip (NoC), and leverages mini-batch spiking Gustavson-product to cut memory access and exploit inherent sparsity. Combined with mapping and scheduling optimizations, ELSA achieves efficient, event-driven computation without compromising accuracy. Experiments show that SNNs can outperform quantized artificial neural networks (QANNs) while maintaining on-par accuracy. For a 4-bit ResNet-50, ELSA achieves 3.4$\times$ speedup and 13.6$\times$ higher energy efficiency over the SOTA QANN accelerator (ANT), and 2.9$\times$ speedup and 22.1$\times$ energy efficiency gains over the SOTA SNN accelerator (PAICORE).