Strengthened glass is used for display applications because of its high resistance to scratch damage. The high strength is the result of an ion exchange, which leads to a change of the glass structure and therefore the range of compressive stresses in the glass surface increases. These stresses make laser processing of this type of glass more difficult, where chipping and cracking are more likely to occur in comparison to non-strengthened glass. In this study, laser ablation of chemically strengthened glass (Gorilla® Glass 3) is investigated with a picosecond laser. The aim of this investigation is to find process parameters which allow drilling through a 1.1 mm thick glass substrate. The study starts with laser ablation experiments on the glass surface. In this context, the laser fluence, the repetition rate and the pulse overlap are varied to realize high material removal and to minimize chipping. The ablation efficiency was investigated as a function of the laser fluence, whereas chipping and cracking were strongly dependent on the pulse overlap. The best parameters were accepted to generate a drill hole with 5 mm diameter. In order to realize a reliable drilling process, it was necessary to remove the ion exchanged areas on both glass surfaces before drilling through the bulk material. After removing the strengthened glass areas, laser processing was possible at higher average laser power without decreasing the drill hole quality.