Word for 'when someone perceives life as a game'

You've already used a form of the right word—they are "gamifying" their life:

to make an activity more like a game in order to make it more interesting or enjoyable:
Is gamifying the news the future?

Usually we see it used as "gamification", but "gamify" is valid. Forbes uses "gamify" in the title of an article, and includes "Gamifying the workplace" as a section within it:

Gamify Your Life and Become Massively Successful

I can't think of a good noun for this, but I wonder if some kind of metaphor or simile might work?

e.g. "The world is her Playstation, and it's game over for anyone in the way", "If the street were a screen, he'd be Pacman", "Each day was a like new level of Super Mario Bros, harder than the last, yet he had always found his princess by sunset"....

(I'm out of touch with the culture, so forgive the dated ideas!)

Not a noun but for such kind of person, I would use flippant or frivolous.

The action of perceiving life as a game could be described as frivolity.

Frivolity: behaviour that is silly and not serious, or things that are silly and not important. [Cambridge English dictionary]

Example: They should not treat their life with frivolity.

In your example, you could say:

If someone thinks that they are speedrunning a task in life, they do it frivolously.

Light-minded or skittish are also possible. (Is it self-gamification?)