Yielding properties of Au point contacts of nanometer-scale dimensions have been studied using a scanning tunneling microscope supplemented by a force sensor for measuring tip-sample forces. The contacts are made by indenting the tip typically 10 nm into the substrate, whereby an adhesion neck is formed. Three consecutive deformation phases of the neck can be identified during retraction of the tip : (1) buildup of tensile stress, (2) incomplete fracture, and (3) quasicontinuous plastic flow. Finally the neck breaks when a maximum of three to four atoms are left in the contact. In the plastic flow regime, the conductance and thus the contact area shrink exponentially with elongation of the neck. This provides strong evidence that plastic deformation occurs locally within five to six atomic layers. The latter is inferred from the decay constant of the exponential decrease of the cross section. The applied stress during plastic flow is initially of the order of 10 GPa and gradually increases to similar or equal to 20 GPa shortly before the neck breaks. From a fit to the data accounting for a surface force contribution an intrinsic yield strength of the order of 6 GPa is obtained, which is more than one order of magnitude larger than the macroscopic yield strength of Au.