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.