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Our paper is devoted to the study of the holonomy groups of Finsler surfacesusing the methods of infinite dimensional Lie theory. The notion ofinfinitesimal holonomy algebra will be introduced, by the smallest Lie algebraof vector fields on an indicatrix, containing the curvature vector fields andtheir horizontal covariant derivatives with respect to the Berwald connection.We obtain that the topological closure of the holonomy group contains theexponential image of any tangent Lie algebra of the holonomy group. A class ofRanders surfaces is determined, for which the infinitesimal holonomy algebracoincides with the curvature algebra. We prove that for all projectively flatRanders surfaces of non-zero constant flag curvature the infinitesimal holonomyalgebra has infinite dimension and hence the holonomy group cannot be a Liegroup of finite dimension. Finally, in the case of the Funk metric we provethat the infinitesimal holonomy algebra is a dense subalgebra of the Liealgebra of the full diffeomorphism group and hence the topological closure ofthe holonomy group is the orientation preserving diffeomorphism group of thecircle.

Finsler spaces with infinite dimensional holonomy group