Difference between relativistic and non-relativistic classical equations of electron

Both in classical (non-relativistic) electromagnetism and in relativity the equation of motion of an electron in electromagnetic field is
$$ frac{dmathbb{p}}{dt}= e(mathbb{E}+frac{mathbb{v}}{c}times mathbb{B}).$$
The momentum $mathbb{p}$ in the non-relativistic situation is $mmathbb{v}$, while in the relativistic one it is $frac{m}{sqrt{1-v^2/c^2}} mathbb{v}$. Here $m$ is the rest mass of the electron in both equations.

I am wondering whether the difference between two equations has been detected experimentally for high speed electrons. (I do not know if that it practically doable due to possible presence of quantum effects.)