What is highest water pressure at which electrolysis can be performed to derive hydrogen and oxygen? Does the dielectric constant of water, which which lowers as pressure increases, have an effect on this?
A high-pressure environment tends to favor liquids over gases, insofar as liquids have higher density. So the higher pressure it is, the higher voltage you need to apply to make the reaction H2O --> H2+O2 occur.
But, with a high enough voltage, it should always be possible. ANY electrolytic reaction will eventually occur if you crank up the voltage high enough. Electrolysis only becomes impossible (in practice) (impossible at any voltage), if competing reactions are favored. I'm not sure what the competing reactions would be for electrolyzing pure water in a high-pressure environment. For example, presumably, H2O --> H2 + O3 (ozone gas) would be favored over H2O --> H2 + O2 at high enough pressure.
It's impossible to be quantitative here, because the catalyst and reaction mechanism also make a huge difference to favor one reaction over another. It's not enough to JUST calculate free energies.