Early instances of 'play out of one's skin' in the UK
The exact expression "play out of [one's] skin" is at least 47 years old and may be somewhat older than that. The earliest match I've been able to find (without access to British newspaper archives), is from 1976. From Ken Jones & Pat Welton, Soccer Skills & Tactics (1976) [snippet view]:
Although Wembley's famous turf had been abused beyond recognition, Leeds set out to achieve a colourful victory. They would have done so had not Bonetti played out of his skin throughout two hours of gruelling action.
A New Zealand bookseller asserts that this book was originally published in 1973 as The Professionals' Book of Skills & Tactics, although it may date back even farther—to 1971, in an edition published by Marshall Cavendish in England; unfortunately, I can't confirm whether the "played out of his skin" idiom appears in those earlier editions of the book.
The psychological aspect of this expression is notable from a fairly early date. From Angela Patmore, Sportsmen Under Stress (1986):
Naturally he has to swing the club and hit the ball. Naturally he has to hole putts, and the way in which he does these things is based on hours and hours of grooved technical perfection. But as he goes from tee to green in a championship, these skills change. They may desert him altogether, or they may eerily seem to produce themselves (called 'playing out of one's skin') so that the player can apparently do no wrong. As he progresses across the hills and hazards, with thousands of pairs of eyes glued to his every move, the professional golfer knows that they are not simply watching where he puts his hands on the club, how he swings, or even where the ball goes. He knows that they are watching the journey in his mind. They want to know what happens when he comes across this or that emotion. They want to see if he can beat his inner self; if he cannot do that, the bunkers will get him, or the Road Hole will get him, or he will 'take the gas' on the 18th green. And because the golfer has more time to consider this ...
Golfers call their own mediumship 'playing out of your skin' or 'playing like God's professional'. They freely admit that it is not something they can control, although many have tried. They talk of their 'visions' of perfect shots. Arnie Palmer: 'But something else happened that is harder to explain – and yet is vital to continued great play whenever trouble looms. That something was visual, not intellectual; I could literally see the shot that had to be made.'
A Google Books search turns up two matches from 1988, one in connection with sport and the other with jazz music.
From Charles Benson, No Regard for Money: The Memoirs of a Racing Man (1988) [combined snippets]:
Then, amidst incredible excitement from the crowd, Botham went mad and scored one of the mot amazing centuries in cricket history. He didn't stop at a 100, and by the end of the innings he had helped England to a lead of 129, still barely enough in normal circumstances. But then it was the turn of Botham's friend, Bob Willis to play out of his skin and he took eight wickets to enable England to register an incredible victory. Dennis could not wait to tell me about his amazing 240.
And from an unidentified item from in Timber Trades Journal & Wood Processing, volume 347 (1988):
And the special attraction will be (no, not Paul in his bookie's hat) but The Smokey City Jazz Band, who played out of their skin last year.
From an unidentified article in in New Statesman Society (1989):
At Ansfield, the general opinion is that Gayle had a chip on his shoulder—"he was a bit, you know, black power", says one observer. Barnes, on the other hand, is everybody's favourite, an aristocrat capable of playing "out of his skin", as the jargon has it . His skills are easily the best response to the venomous Derby-day chant "Everton are white! Everton are white!"
From Michael Globetti, God Save the Quarterback!: American Football Goes to England (1991) [combined snippets]:
But filling in, the team instantly uplifted by him, Jeffo got off to an astonishing start. A few days earlier, giving in to the hypochondriacal litany that often kept him out of practice, he had left Moor Lane complaining of chest pains. "Probably something with my diaphragm," he groaned. "The old man used to be top medical officer in the British army, and that's what he says, anyhow." Now he was bucking us up big time. "Hike!" with a British accent had never sounded so good, and he knew what to do when the football arrived in his tiny hands, too. Never having made more than a single completion in a game , he played out of his skin, and out of his 5-foot-5 stature, not only admitting to no pancreatic problems on the pitch, but making good on six of his first eight passes, everything on target. It was as if Anne Boleyn were looking down with her six-fingered hands cupped over him.
From an unidentified article in Jazz Journal International volume 46 (1993) [combined snippets]:
This was well illustrated recently when my wife and I attended one of Wigan Jazz Club's Sunday lunchtime gigs at The Mill At The Pier. The featured musician was a much slimmed down Don Weller (can the be a better tenor player in Britain at present?) He fronted a local pack-up trio led by a stalwart of NW Jazz, Joe Palin. Joe, the young bass player and even younger drummer played out of their skin (literally in the case of the bass player!) to give Don a firm foundation on which to build his solos. Don's reading of Soul Eyes was worth the admission on its own.
A related phrase "racing out of their skin" appears in "Ovett Writes Off Out-of-Form Cram," in the Canberra [ACT] Times (August 6, 1987):
"When you look at our team, you have to think we could return from a championship for the first time in a decade without a gold at 1500m," Ovett said, who has only been chosen for the 5,000m.
"No one is racing out of their skin right now. We're all having an indifferent season.
"Ovett" is Steve Ovett, an elite middle-distance runner from the UK. The article is datelined London.
Early popularization of 'play out of one's mind' in the U.S.
Interestingly, in U.S. sports slang the term "play out off one's mind" seems to be about the same age as the British "play out of one's skin" and carries very nearly the same meaning. The popularization of "play out of one's mind," however, may be linked to W, Timothy Gallwey's best-selling book The Inner Game of Tennis (1974):
Listen to the phrases commonly used to describe a player at his best: "He's out of his mind"; "He's playing over his head"; "He's unconscious"; "He doesn't know what he's doing." The common factor in each of these descriptions is what might be called "mindlessness." There seems to be an intuitive sense that the mind is transcended—or at least in part rendered inoperative. Athletes in most sports use similar phrases, and the best of them know that their peak performance never comes when they're thinking about it.
Clearly, to play unconsciously does not mean to play without consciousness. That would be quite difficult! In fact, someone playing "out of his mind" is more aware of the ball, the court, and, when necessary, his opponent. But he is not aware of giving him- self a lot of instructions, thinking about how to hit the ball, how to correct past mistakes or how to repeat what he just did. He is conscious, but not thinking, not over-trying. A player in this state knows where he wants the ball to go, but he doesn't have to "try hard" to send it there. It just seems to happen—and often with more accuracy then he could have hoped for. The player seems to be immersed in a flow of action which requires his energy, yet results in greater power and accuracy. The "hot streak" usually continues until he starts thinking about it and tries to maintain it; as soon as he attempts to exercise control, he loses it.
I don't know whether there is a direct link in lineage between the U.S. "play out of one's mind" and the British "play out of one's skin," but it wouldn't shock me if there were.
Prior sporting use of 'jump [or leap] out of one's skin'
Another (and much older) idiomatic expression that may bear on the emergence of "play out of one's skin" is "jump [or leap] out of one's skin." Although people commonly use this expression in connection with sudden fright, it also occurs in the context of elation or boundless energy. For example, from "Power and Thomas Reach Final," in the Canberra [ACT] Times (September 2, 1960):
"We have just arrived here really, and we walk into the Olympic Games. The Germans have come from an obviously terrific preparation on European tracks."
[Friedrich] Janks was jumping out of his skin as he finished two seconds ahead of Thomas in the fastest of the four heats.
And from"Rose Favoured for Fight," in the Canberra [ACT] Times (August 16, 1968):
Rennie said Rose spent two hours at the punchbag and skipping at a gymnasium prepared for him at a Los Angeles hotel.
"In a fortnight's time he should be jumping out of his skin. He is very keen to impress here.
"Lionel is starting favourite, being the champion, but we realise this is going to be a difficult one".
As is the case with "play out of one's mind," the possibility that "jump out of one's skin" influenced the emergence of "play out of one's skin" is neither certain nor (it seems to me) far-fetched. The expression "play out of one's skin" was definitely in use by 1976 in the UK, although it may not have achieved widespread adoption there until the late 1980s.