In core Savage Worlds, characters can indeed die. The most common way is through Incapacitation.
Wild Cards are Incapacitated if they suffer more than three wounds (cumulatively or all at once). When a Wild Card becomes Incapacitated, make an immediate Vigor roll:
• Total of 1 or Less: The character dies.
–Savage Worlds Deluxe, Chapter Three: Game Rules, "Damage Effects"
There is also a result of "Bleeding Out", which means that in future rounds, further rolls will be attempted to see if the character dies.
However, there is an optional setting rule called "Heroes Never Die" that prevents death except during thematically appropriate events.
Heroes in movies very rarely die. And when they do, they go down fighting or perform one last, epic act of heroism.
With this rule in play, heroes who would normally die are simply knocked out for a dramatically appropriate time instead. Only if the situation is particularly heroic, or if it serves as a major story point can characters actually die.
An adventurer who falls into a volcano, for example, might land on a oating piece of rock and leapfrog to safety. Conversely, if a hero confronts a massive demon on a crumbling bridge and suffers what would normally be a mortal wound, perhaps he does perish but the Game Master decides with his final blow he takes the demon into the boiling lava with him, allowing his companions to escape.
–Savage Worlds Deluxe, Chapter Four: Situational Rules, "Setting Rules"
There are some published settings that follow the Heroes Never Die rule, or some variation of it (e.g. Daring Tales of Adventure).
Note that unlike your GM's rule, Heroes Never Die doesn't have the character come back with a Hindrance. It's likely that your GM was either following a rule included with a published setting, or they houseruled something.