In the Alps to the north, one finds Alpine chamois, which is open to hunting; in the central part on the Apennine there is the Abruzzi chamois (which is closed to hunting). Also in the Alpine north, there are Alpine ibex, which are closed to hunting. In the north and central part of the country there is a population of unique red deer that has been confirmed by genetic and systematic research as a special subspecies called Cervus elaphus italicus (Italian red deer), which differs from the central European deer (Cervus elaphus elaphus) and is open to hunting. There are also many roe deer, mouflon sheep and most of all, wild boar, whose population has exploded in number and range in the last 20 years, covering the whole Italian territory.
Moving off the mainland to the big islands of Sardinia and Sicily, there is an autochthonous Corsican mouflon that is closed to hunting and an indigenous red deer very different from the continental species and similar to the Spanish red deer. This species is also closed to hunting.
On the smaller islands, there is a famous species of feral goat called the Monte Christo, also prohibited to hunt. In the central part of Italy there is wolf (closed to hunting) and small isolated populations of the autochthonous subspecies of Apennine bear (Ursus arctos marsicanus), and as you've probably guessed, it is also closed to hunting.