Technically, Sanger Rainsford never really escapes the island in "The Most Dangerous Game," by Richard Connell. He asks to leave on the same night he arrives and as soon as General Zaroff reveals that he intends to hunt Rainsford; however, Zaroff refuses his request.
After three days of being hunted, Rainsford finds himself trapped. Zaroff and his dogs are behind him, and Rainsford has very little choice about his next move.
A blue gap showed between the trees dead ahead. Ever nearer drew the hounds. Rainsford forced himself on toward that gap. He reached it. It was the shore of the sea. Across a cove he could see the gloomy gray stone of the chateau. Twenty feet below him the sea rumbled and hissed. Rainsford hesitated. He heard the hounds. Then he leaped far out into the sea....
This, then, is how he "escaped" from the island, but of course he returns. That night, a contented Zaroff goes to his bedroom and turns the light on; he is shocked to find Rainsford hiding behind his curtains.
"Rainsford!" screamed the general. "How in God's name did you get here?"
"Swam," said Rainsford. "I found it quicker than walking through the jungle."
That short time between jumping off the cliff and returning to Zaroff's mansion is Rainsford's only "escape" from the island. We can assume, however, that he eventually does leave Ship-Trap Island--but not until after he has a good night's sleep in General Zaroff's bed.