An invert emulsion drilling fluid comprises: an external phase, wherein the external phase of the drilling fluid comprises a hydrocarbon liquid; an internal phase, wherein the internal phase of the drilling fluid comprises a hygroscopic liquid; and a suspending agent, wherein the suspending agent is a polymer comprising urea linkages or urea and urethane linkages. The hygroscopic liquid comprises a salt and a suitable solvent or comprises an alcohol. The drilling fluid can exclude an organophilic clay or organophilic lignite. A method of using the invert emulsion drilling fluid comprises: introducing the drilling fluid into at least a portion of a subterranean formation.