The paper is devoted to the statistical analysis of the end- to-end (E2E) delay of packet transfers between source and destination nodes in a peer-to-peer (P2P) overlay network. We focus on the iden- tication of the E2E delay and the longest per-hop delay distributions and the stochastic dependence of the associated random process. The E2E delay is determined by the sum of a random number of dependent per-hop (p-h) delays along the links of a considered overlay path and the longest per-hop delay by their maximum. We propose to use the sum of the p-h delays to get a distribution of the maximum which is motivated by the available statistical data of the E2E delays. Based on recent ana- lytic results derived from extreme-value theory we show that such sums and maxima corresponding to different paths may have the same tail and extremal indexes. These indexes determine the heaviness of the dis- tribution tail and the dependence of extremes. Using the extremal index we identify limit distributions of the maxima of the E2E delays and the maxima of the p-h delays at a path among all source-destination paths. Considering real-time applications with stringent E2E-delay constraints, the distributions are used to identify quality-of-service (QoS) metrics of a P2P model like the packet missing probability and the corresponding playback delay as well as the equivalent capacity of a transport channel.