What you are looking for are called *perfect nonlinear* or differentially $1-$uniform functions.

They don't exist over even characteristic since if $x_0$ satisfies
$$
f(x+a)-f(x)=b,
$$
so does $x_0+a.$

For a long time only some power functions or functions equivalent to them were known. A recent paper lists
the following known examples among others.

$$x^2~~ in ~~GF(p^n), $$

$$x^{p^k+1} ~~in ~~GF(p^n),\quad k \leq n/2~~and ~~n/(k,n)~~odd$$

$$x^{10} + x^6 − x^2 ~~in~~ GF(3^n), ~~n \geq 5 ~~odd$$

See *New families of perfect nonlinear polynomial functions* by
Zhengbang Zha, Xueli Wang, Journal of Algebra (322):3912-3918.

for more. One class of such polynomials are called Dembowski-Ostrom polynomials. All of the above are, but a recent example $$x^{(3^k+1 )/2}$$ isn't.

**Edit:** *I apologise, I shouldn't post before my morning coffee. This is now essentially a long comment.*

*The functions displayed are not involutions. Recent work on involutions is [here][1]. I think your question is quite difficult.*


  [1]: https://arxiv.org/abs/1811.11344