Stretchable electronics find widespread uses in a variety of applications such as wearable electronics, on-skin electronics, soft robotics and bioelectronics. Stretchable electronic devices conventionally built with elastomeric thin films show a lack of permeability, which not only impedes wearing comfort and creates skin inflammation over long-term wearing but also limits the design form factors of device integration in the vertical direction. Here, we report a stretchable conductor that is fabricated by simply coating or printing liquid metal onto an electrospun elastomeric fibre mat. We call this stretchable conductor a liquid-metal fibre mat. Liquid metal hanging among the elastomeric fibres self-organizes into a laterally mesh-like and vertically buckled structure, which offers simultaneously high permeability, stretchability, conductivity and electrical stability. Furthermore, the liquid-metal fibre mat shows good biocompatibility and smart adaptiveness to omnidirectional stretching over 1,800% strain. We demonstrate the use of a liquid-metal fibre mat as a building block to realize highly permeable, multifunctional monolithic stretchable electronics.