He left the pitch with the match ball (in hand) after scoring a hat-trick.

After the game, he confirmed he'd managed to get his hands on a match ball to mark his achievement.
http://www.express.co.uk/sport/football/526974/Harry-Kane-ma...

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Traditionally, the scorer of a hat-trick is entitled to take home the match ball.
http://news.bbc.co.uk/sport1/hi/football/teams/r/rushden_and...

https://www.quora.com/If-more-than-1-player-scores-a-hatrick...
https://www.theguardian.com/football/2004/jan/28/theknowledg...


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Note added at 3 hrs (2017-10-15 13:52:07 GMT)
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Remember when Messi and Suarez both Scored Hatrick against Valencia in 2015?

In that match, Suarez Scored 4 and Messi scored 3 and they won by 7–0 in the Copa Del Rey semi-final.

However, Messi gave Suarez the Match ball after the final whistle was blown. Probably he was also thinking only one gets the ball. xD

But eventually, they both got the ball.

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Note added at 4 hrs (2017-10-15 15:26:46 GMT)
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The tradition of leaving the field with the match ball is a just reward for having bagged three goals in a single game.
http://www.fifa.com/news/y=2011/m=4/news=handful-heroic-hat-...

Mick Jones made the headlines on February 19, 1972 with his first league hat-trick and one of only four ever scored in meetings between the clubs. For no apparent reason, and to his eternal regret, Jones left Elland Road after a 5-1 win without the match ball. “It annoys me that I didn’t take it with me,” he says. “That hat-trick was a real highlight of my career and the match ball is your reward for it.
http://www.yorkshireeveningpost.co.uk/sport/football/leeds-u...

Gareth Bale holds the match ball aloft, his reward for a hat-trick at Villa Park
http://bleacherreport.com/articles/1458469-gareth-bale-best-...