We define the hyperbolic Yamabe flow and obtain some properties of its stationary solutions, namely, hyperbolic Yamabe solitons. We also consider immersed submanifolds as hyperbolic Yamabe solitons and prove that, under certain assumptions, a hyperbolic Yamabe soliton hypersurface is pseudosymmetric or metallic shaped. Further we characterize the hyperbolic Yamabe soliton factor manifolds of a multiply twisted, multiply warped, doubly warped, and warped product manifold, and we provide a classification for a complete gradient hyperbolic Yamabe soliton factor manifold. Finally we determine the conditions when the factor manifolds are hyperbolic Yamabe solitons if the manifold is a hyperbolic Yamabe soliton and illustrate this result for the Robertson--Walker spacetime.