Can you make Bearnaise with olive oil?

Don't make any emulsion sauces with olive oil. The high speed of dispersion, especially with an electric blender, creates unpleasant bitter compounds in the oil. But frequently, even the speed of a hand whisk is high enough to ruin the taste. Either that, or you are too slow to create the emulsion at all.

You can form yolk-fat-emulsion sauces with practically any fat, and olive oil is an unfortunate exception with its taste change. Just choose another fat compatible with your diet. From a purist's point of view, they won't be a Bearnaise, the French sauces are very tightly classified. You will also get an obvious difference in taste (only butter tastes like butter), and, if you are using a liquid oil, a slight difference in handling - you can let a liquid-oil-emulsion cool down, while a butter or lard based sauce will harden into an unpleasant grainy mass.

If your base food is very plain/bland tasting, you can play with subtle oil tastes to create an interesting effect where the main "seasoning" is the oil in the sauce.

Yes, you can make a "light Bearnaise" with olive oil, in fact Gordon Ramsay has done it before and it's featured in a recipe somewhere. Just warm the oil slightly (as you would warm the ghee/clarified butter in an actual Bearnaise sauce,) to keep the temperature correct.

I usually make it this way, as I have more olive oil than I do ghee, also the smell of ghee gets to me a bit. The sauce comes out lighter and you can taste the sourness a bit more, so it would be advisable to ease up on the vinegar a bit.

Since most Bearnaise recipes use a volumetric amount of ghee, you can simply substitute for the same volumetric amount of olive oil (which you may or may not use all of, since you have to judge by eye whether the mixture is about to split or not.)

Source: me. Also Gordon Ramsay did it.

It wouldn't be bearnaise then. Can you make hollandaise with olive oil? It would taste so different that it wouldn't be hollandaise.