Top-k query processing in P2P systems has focused on efficiently computing the top-k results while reducing network traffic and query response time. However, in overloaded P2P systems (with very high query loads), some peers may take a long time to answer, thus making the user wait a long time to obtain the final top-k result. In this paper, we address this problem, which we reformulate as early top-k query processing in P2P systems. First, to complement response time, we introduce two new metrics, stabilization time and cumulative quality gap, with which we formally define the problem. Then, we propose an efficient algorithm that dynamically adapts to query loads of peers in order to return to the user top-k results as soon as possible, without waiting for the final result. We validated our solution through simulations over a real dataset. The results show that our solution significantly outperforms baseline solutions by returning high quality top-k results to users in much better times.