Summary: A higher dimensional analogue of the dispersionless KP hierarchy is introduced. In addition to the two-dimensional “phase space” variables \((k,x)\) of the dispersionless KP hierarchy, this hierarchy has extra spatial dimensions compactified to a two (or any even) dimensional torus. Integrability of this hierarchy and the existence of an infinite dimensional of “additional symmetries” are ensured by an underlying twistor theoretical structure (or a nonlinear Riemann-Hilbert problem). An analogue of the tau function, whose logarithm gives the \(F\) function (“free energy” or “prepotential” in the context of matrix models and topological conformal field theories), is constructed. The infinite dimensional symmetries can be extended to this tau (or \(F)\) function. The extended symmetries, just like those of the dispersionless KP hierarchy, obey an anomalous commutation relations.