A new type of lyotropic liquid crystalline elastomer is described. The elastomer is synthesized by cross-linking a lyotropic epoxide-amine addition polymer that exhibits a hexagonal phase in the binary polymer water mixture. A similar phase behavior is revealed in the water-swollen elastomer. Uniaxial strain of the network causes a macroscopically uniform orientation of the rodlike micelles parallel to the stress axis. Such an oriented sample is characterized by hygroelastic measurements at different temperatures and uniaxial strains monitoring the network dimensions as a function of the water content. Because of the coupling of the lyotropic liquid crystalline phase structure and the network chain conformation, the network swells anisotropically within the hexagonal phase. At the isotropic to hexagonal phase transformation a discontinuous change in length is observed. A partial protonation of the linear polymers dramatically increases the concentration and temperature regime of the hexagonal phase and additionally the solubility in water. Applying this knowledge to the cross-linked system, switching between isotropic and the hexagonal phase is realized by variation of the pH value. The phase transformation becomes apparent in the anisotropic change of dimensions of the network.