Well, the first example that I can think is when Rainsford is standing on the deck of the yacht at night and he hears a gunshot. Trying to get a clearer look, Rainsford loses his balance and falls into the water. Rainsford doesn't try to swim back to the yacht (which is smart, as it would have been futile). The narrator states that "a certain coolheadedness had come to him; it was not the first time he had been in a tight place". Because he was able to stay calm, he was able to swim to Ship Trap Island.
Later on, after General Zaroff had released Rainsford into the jungle, Rainsford kept telling himself repeatably "I must keep my nerve, I must keep my nerve". One final example is how Rainsford is able to keep calm during the hunt and think of ways to trick Zaroff. "An idea that held a wild chance came to him, and, tightening his belt, he headed away from the swamp".