Datacenter networking is currently dominated by two major trends. One aims toward lossless, flat layer-2 fabrics based on Converged Enhanced Ethernet or InfiniBand, with benefits in efficiency and performance. The other targets flexibility based on Software Defined Networking, which enables Overlay Virtual Networking. Although clearly complementary, these trends also exhibit some conflicts: In contrast to physical fabrics, which avoid packet drops by means of flow control, practically all current virtual networks are lossy. We quantify these losses for several common combinations of hypervisors and virtual switches, and show their detrimental effect on application performance. Moreover, we propose a zero-loss Overlay Virtual Network (zOVN) designed to reduce the query and flow completion time of latency-sensitive datacenter applications. We describe its architecture and detail the design of its key component, the zVALE lossless virtual switch. As proof of concept, we implemented a zOVN prototype and benchmark it with Partition-Aggregate in two testbeds, achieving an up to 15-fold reduction of the mean completion time with three widespread TCP versions. For larger-scale validation and deeper introspection into zOVN, we developed an OMNeT++ model for accurate cross-layer simulations of a virtualized datacenter, which confirm the validity of our results.