What is the story about Merlin, which includes Gorlois' captain Brithael?
In the story, Merlin has put a glamour on Uther, which allows Uther to impersonate Gorlois, who is off at war, and impregnate Gorlois' young wife, Ygraine. One sentence reads, approximately, "A justly enraged Brithael stomped on my fingers" as Merlin is clinging to a precipice.
I suspect you're thinking of The Crystal Cave by Mary Stewart, but the fight between Merlin and Brithael is not as you remember:
He was a big man, a fighter in his prime, and justly angry. I was without experience, and hating what I must do, but I must do it. I was no longer a prince, or even a man fighting by the rules of men. I was a wild animal fighting to kill because it must.
...
But before even the sound reached us he was on me again. I had a knee under me and was dragging myself up painfully. Through the blood in my eyes I saw the blow coming, and tried to dodge, but his fist struck me in the throat, knocking me sideways with a savagery that spread-eagled me again on the wet turf with the breath gone from my body and the sight from my eyes. I felt myself roll and slip and, remembering what lay below, blindly drove my left hand into the turf to stop myself falling. My sword was still in my right hand. He jumped for me again, and with all the weight of his big body brought both feet down on my hand where it grasped the sword. The hand broke across the metal guard. I heard it go.
It does have the justly angry phrase, and Brithael does stamp on Merlin's hand, so it could be your book if you have misremembered the scene a bit.
The fight does happen at the top of a cliff. A few paragraphs later we read:
He came in again as quickly as before, and even as I tried to drag myself away, he made a quick stride forward and stamped again on my broken hand. Somebody screamed. I felt myself thrash over, mindless with pain, blind. With the last strength I had I jabbed the sword, hopelessly shortened, up at his straddled body, felt it torn from my hand, and then lay waiting, without resistance, for the last kick in my side that would send me over the cliff.
But Merlin is not hanging from the cliff by his fingers when Brithael stamps on them (or indeed at any other time). Merlin's blind thrust with the sword has fatally wounded Brithael and Merlin just lies there at the top of the cliff until he has recovered enough to stand up.
