Merriam-Webster's Eleventh Collegiate Dictionary (2003) dates the adjective game in the sense of "having or showing a resolute unyielding spirit" to 1610. That being the case, it's a bit surprising that a Google Books search doesn't turn up any relevant seventeenth- or eighteenth-century matches for "I am game," "you are game," we are game," "they are game." he is game," or "she is game." Accurate though MW's definition and dating may be for the adjective game, the phrase "I am game" (or "he is game," etc.) may have arisen as late as the early nineteenth century.
Fans of Dickens's Dombey and Son (1846–1848) may recall a character whose role in the book is to teach Mr. Toots (an amiable but rather gullible young gentleman) various manly arts—a character identified only as "the Game Chicken." As you may know, Dickens named this character after a champion pugilist of the early nineteenth century, one Henry Pearce (aka "Hen" Pearce, aka the Game Chicken). Pearce died in 1809 just before the age of 33, but he was England's bare knuckles prizefighting champion between 1804 and 1807.
I was delighted to learn in John Farmer & William Henley, Slang and Its Analogues (1893) that some word historians credit the Game Chicken with vitalizing the term game:
Game, ... Adj. (old).—1. Plucky ; enduring ; full of spirit and BOTTOM (q.v.). (Cock-pit and pugilists'. The word may be said to have passed into the language with the renown of Harry Pearce, surnamed the GAME CHICKEN.]
Pearce was certainly was a very famous figure in his day (and afterward). Though the Wikipedia article on Pearce claims that he earned the sobriquet "the Game Chicken" by referring to himself as "Hen," his obituary in the June 1809 issue of The Athenæum indicates that his tenacity was responsible for the Game part of the name:
Pearce first entered the lists with Bourke, whom Belcher had twice beaten, and they fought in a room in St. Martin's-lane by candle-light. The conflict was short and desperate, and in a quarter of an hour the Bristol hero was declared the victor. The bottom he evinced on this occasion procured him the name of Game Chicken ; upon which he crowed defiance of all the game cocks in the kingdom, Belcher excepted (it being his intention not to pit himself against any of the Bristol breed.)
The crucial word in this excerpt is bottom, which Eric Partridge, A Dictionary of Slang and Unconventional English, Eighth edition (1984) defines as "Stamina, grit," citing Captain Godfrey, The Science of Defence (1747) to this effect:
Bottom, that is, wind and spirit, or heart, or wherever you can fix the residence of courage...
Farmer & Henley sites Captain Godfrey as saying elsewhere in the same book,
Smallwood [a boxer] is thorough game, with judgment equal to any, and superior to most.
In any event, the earliest instances of I am/he is/they are/... game that a Google Books search finds are from the 1800s. From Byron, "Don Juan," Canto VII (July 15, 1823):
The second object was to profit by
The moment of the general consternation
To attack the Turks' flotilla, which lay nigh
Extremely tranquil, anchor'd at its station:
But a third motive was as probably
To frighten them into capitulation ;
A fantasy which sometimes seizes warriors,
Unless they are game as bull-dogs and fox-terriers.
From American Turf Register (December 1832):
As for Black Maria, she is literally "too fast for the speedy, and too strong for the stout." She ran the twentieth mile with a freshness and vigour that surprised every body, and the spectators at last actually conceded that she is "game!" That she can conquer either Relief or Trifle, at two heats, in a match, there can be no manner of doubt; and that she is a "hard one to beat" in any race, even by a field, all sportsmen must now believe.
From Curio, "The Sanguine Man," in The Metropolitan, volume 36 (1843):
The lessons of life lighten, but not enlighten him. "The uses of adversity" to him are rather bracing than "sweet," they give him nerve without philosophy, plasticity without power, aspirations in place of honour, multifold "kicks" and very few "halfpence;" and hapless as his case must be with the genius he defies, he is game and dilemma-proof to the last, and comes off "more than a conqueror," though a very considerable loser, from every conflict in which he shrinks not from engaging.
From Albert Smith, "The Scattergood Family," in Bentley's Miscellaney (1844):
"If I can get out of this, I'm game," returned Bolt.
"If you are game, they'll bring you down — safe," answered the other, with a chuckle.