Plumbing – How to detach this faucet sprayer? Photos included

Your sprayer unit is not designed to sit at full system pressure, so the faucet unit that it came with is specially designed to shunt water towards the sprayer hose and head only when the sprayer lever is pressed. The faucet itself has an integral diverter valve to perform this function, it sounds like yours is not working right.

All faucet brands use their own proprietary diverter mechanism, you will need to get a manual for your faucet to get it apart and replace the valve. It might look something like this:

The easiest option in my opinion is to get a sink with a 4th hole cutout for the soap dispenser if you want that. They are a fairly standard option when searching for sinks and I think the resulting look is the best.

It may be possible to drill another hole, but I wouldn't recommend it. If you attempt this, please use a center punch to make a dent to start the hole saw in, otherwise the bit will slide all over the sink and not where you want to drill. The reason I wouldn't consider it is the high risk of damaging the finish and possibly making the hole in the wrong place.

If you don't like the first two options, you can only use one hole in the sink for most modern faucets. They typically include an option to install a cover plate to hide the other two holes. If you don't use that plate, you have two holes to use for soap and lotion dispensers.