If the tap was fitted properly then there should be an isolation valve to turn off underneath so that you can have the rest of the water on. If not then fit one or two.
You can get new tap inners if you know the make of tap and where it came from. This can be a mess about and often it means getting a new tap.
If you can isolate the tap you could have a play about with it and strip it down It sometimes works but dont count on it. At least once you have the faulty part in your hand you can take it to plumbers merchants and BnFnQ etc and see about a replacement.
I came on here to post the exact same question. Except my mixer taps are metal not ceramic.
First of all I attacked the hotwater tap itself removing the tap and the cover. So i got down to the spindle and I managed to prise off the little circlip between it and the base of the mixer main body. Thats where i am now as i cant remove the rest of it to get at the o rings or any washers.
It appears to be screwed into the mixer and i dont have any spanners to fit it. I've tried with pliers but they have failed me miserably. I think its stuck fast as the metal started to disolve under the pressure of my pliers grip.
Also even after removing the tiny screw of the main spout section it will not come off either. It swivils around full 360 degrees to infinity but doesnt screw off. I tried pulling it up vertically but no joy either.
Its a B&Q mixer i had installed with my new kitchen about 3 years ago so it shouldnt be in that state already?
Looks like I'm going to have to make that call for a plumber but as I really am in financial crud at the moment so i thought I'd ask on here first.
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