  The affine Gaudin model, associated with an untwisted affine Kac-Moody
algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed
topological-holomorphic Chern-Simons theory in the Hamiltonian framework. We
show that the finite Gaudin model, associated with a finite-dimensional
semisimple Lie algebra, or more generally the tamely ramified Hitchin system on
an arbitrary Riemann surface, can likewise be obtained from a similar gauge
fixing of 3-dimensional mixed BF theory in the Hamiltonian framework.
