Is it possible to have electric field in water without having electrolysis?
Well, if the voltage is lower than needed for any possible electrolysis reaction, there would be no reaction, just the field. So the answer may seem to be yes.

Now, in fact there is a catch. Once you turn on the voltage, even if the reaction is not possible, the ions will rush to the respective electrodes and form the so-called electric double layer. Its thickness, known as Debye length, is usually quite small. As for the rest of the solution, it will be effectively shielded from the electric field.

Whether or not this counts as a positive answer is up to you.