It is generally characteristic of a field theory with a zero-mass particle that it does not possess a nontrivial Galilean limit. Since all the well-known gauge theories require (at least in the free-field limit) such massless excitations, there are no known examples at this time of Galilean-invariant gauge field theories. However, by making use of a recently formulated gauge theory in two spatial dimensions in which there is no elementary photon, it is shown that there does exist a Galilean theory which incorporates the gauge principle. The general N-particle state for this theory is constructed and subsequently used to obtain the corresponding N-particle Schroedinger equation. In the case of two particles the scattering process is considered explicitly, it is being shown that for all partial waves one obtains the same nonzero phase shift.