It seems to me that this example, from Catherine Mason, The Fisher's Daughter, or The Wanderings of Wolf (1835), envisions springing (albeit metaphorical springing) as an ongoing act of some duration:
I was then wild, little better than a savage boy, untutored, uneducated, still I was steadfast to the truth ; and shall I now forsake that beaten track, which was the light of my existence, and the glory of my heart? Now that I have energy, sensibility, character, now that I feel I am springing into manhood, shall I forsake man's proudest boast—truth? No, let me perish first.
Although a reader might infer that "I am springing" here means "I am about to spring," I think it makes more sense to understand it as "I am in midflight (as I spring from childhood into manhood)."
Similarly, from Philip Doddridge, "Preface," dated December 11, 1746, to volume 8 of The Works of the Rev. P. Doddridge, D.D. (1805):
And if the slender and precarious thread of my life be cut short, before, in the midst of so many other necessary employments, such a work [the completion of three more volumes of writings on religious themes after the publication of two earlier volumes] can be completed, may God graciously accept a purpose with which I trust he has inspired a breast unfeignedly devoted to his service! And may he in that case raise a much abler hand to execute a task, at the prospect of which, though after the preparation of more than twenty years, I feel a secret kind of terror, mingling itself with all the delight with which I am springing forward to undertake it!
Again it's possible to read the springing as not having begun yet, but I see it instead as being a spring already in progress.
A third (and much more recent) instance occurs in Garth Battista, How Running Changed My Life: True Stories of the Power of Running (2002):
Although this is the city, I believe I can smell a river drying. I wiggle my toes. In my body there is the sensation of gentle lifting. I am springing upward inside, as if the weight of snow has fallen from my bough.
In my view, these three examples demonstrate that “I am springing” can indeed have a present continuous meaning relating to a single spring; but the fact that each example can arguably be interpreted as meaning "I am about to spring" suggests why it is so difficult to prove beyond a reasonable doubt that the phrase may refer to an ongoing process of leaping forward or upward.