In this paper, we present SmoothCache, a peer-to-peer live video streaming (P2PLS) system. The novelty of SmoothCache is threefold: i) It is the first P2PLS system that is built to support the relatively-new approach of using HTTP as the transport protocol for live content, ii) The system supports both single and multi-bitrate streaming modes of operation, and iii) In Smoothcache, we make use of recent advances in application-layer dynamic congestion control to manage priorities of transfers according to their urgency. We start by explaining why the HTTP live streaming semantics render many of the existing assumptions used in P2PLS protocols obsolete. Afterwards, we present our design starting with a baseline P2P caching model. We, then, show a number of optimizations related to aspects such as neighborhood management, uploader selection and proactive caching. Finally, we present our evaluation conducted on a real yet instrumented test network. Our results show that we can achieve substantial traffic savings on the source of the stream without major degradation in user experience.