The small punch testing technique under quasi-static loading was examined in the view of punch eccentricity role. It might arise especially when the ball is used as the penetrating tool, instead of hemispherical-ended punch. The austenitic stainless steel AISI 316L was chosen to execute several various “large scale” tests in order to calibrate the multi-linear stress–strain relationship along with the ductile fracture criterion KHPS. All the calibration procedure was performed without using the small punch testing. Then, the model was applied to small punch tests to observe the prediction ability when compared to real small punch experiments. Consequently, a numerical study was conducted to see the role of eccentricity in the case of 2 and 2.5 mm ball diameters used as the penetrating tool. The magnitude of eccentricity up to 0.3 mm was numerically tested. The results showed negligible role of eccentricity for 2 mm ball diameter and minor role for 2.5 mm diameter and studied material.