Nanocomposite networks of surfactant micellar chains and natural bentonite clay nanoplates are studied by rheometry, small-angle neutron scattering, and cryogenic transmission electron microscopy. It is shown that, in an aqueous medium in the presence of a small part of an anionic surfactant, sodium dodecyl sulfate, the molecules of a biodegradable zwitterionic surfactant, oleyl amidopropyl dimethyl carboxybetaine, form micron-length living micellar chains which entangle and form a network possessing well-defined viscoelastic properties. It is found that addition of negatively charged clay nanoplates leads to an increase in viscosity and relaxation time by an order of magnitude. This is explained by the incorporation of the nanoplates into the network as physical multifunctional crosslinks. The incorporation occurs via the attachment of semispherical end-caps of the micelles to the surface of the particles covered with a surfactant layer, as visualized by cryogenic transmission electron microscopy. As the amount of nanoplates is increased, the rheological properties reach plateau; this is associated with the attachment of all end parts of micelles to nanoplates. The developed nanocomposite soft networks based on safe and eco-friendly components are promising for various practical applications.