Offline Imitation Learning (IL) methods such as Behavior Cloning are effective at acquiring complex robotic manipulation skills. However, existing IL-trained policies are confined to executing the task at the same speed as shown in demonstration data. This limits the task throughput of a robotic system, a critical requirement for applications such as industrial automation. In this paper, we introduce and formalize the novel problem of enabling faster-than-demonstration execution of visuomotor policies and identify fundamental challenges in robot dynamics and state-action distribution shifts. We instantiate the key insights as SAIL (Speed Adaptation for Imitation Learning), a full-stack system integrating four tightly-connected components: (1) a consistency-preserving action inference algorithm for smooth motion at high speed, (2) high-fidelity tracking of controller-invariant motion targets, (3) adaptive speed modulation that dynamically adjusts execution speed based on motion complexity, and (4) action scheduling to handle real-world system latencies. Experiments on 12 tasks across simulation and two real, distinct robot platforms show that SAIL achieves up to a 4x speedup over demonstration speed in simulation and up to 3.2x speedup in the real world. Additional detail is available at https://nadunranawaka1.github.io/sail-policy