The only confirmed death is Arrhichion of Phigalia who died whilst winning his 3rd championship. It is difficult to ascertain directly the number of deaths as the historical record simply doesn't (as far as I can see) exist.
That said, we can draw parallels to similar events in more recent history and infer from those how likely death would be in pankration. The largest and most recent is obviously modern MMA, which while it has some restrictions which pankration wouldn't have had, is similar in many aspects.
Officially there have been 16 deaths in MMA - 7 in sanctioned bouts, 9 in unsanctioned bouts, and 0 in the UFC which is the largest promoter. In Vale Tudo - one of the progenitors of modern MMA, the official death record is 0. For MMA this results in about 0.6 deaths/year since UFC 1.
The Greek Olympics ran through about 76 iterations (every 4 years from 776-393BC), so a reasonable assumption if we say it's as dangerous as modern MMA is there would be 57 deaths over the nearly 400 years of the Olympics assuming a similar number of fights/year.
It's not unreasonable to assume fewer deaths than this, however, given that the death rate in bare-knuckle boxing and Vale Tudo are much lower per number of fights than modern MMA. It is well recorded that the introduction of gloves increases risk as head strike frequency increases, so this would definitely change the number down.
When we move on from the Greeks, the brutality of the games increased under the Romans, with the introduction of the Cestus - metal studded leather gloves which essentially weaponised the fist, along with other additions which made the games more lethal.