The Chilean, who arrived from Barcelona for around £32m in July, has made an impressive start to life with the Gunners by scoring seven Premier League goals in 10 games.
However, Walcott, who returned to action in 3-0 win over Burnley after 10 months out, said that Sanchez's teammates have warned him to calm down and preserve some of his energy for hectic festive schedule.
"In training he doesn't even want to do recovery days," Walcott told talkSPORT. "That's how much he loves football.
"All the guys have told him he needs to chill out because at Christmas when all the fixtures come along, it's going to catch up with you.
"He doesn't care, he just wants to play and win. He's a good leader, and something that we haven't had in the Arsenal team before. He's a dream to play with."
Sanchez netted either side of Calum Chambers' first Arsenal goal in the victory over the Clarets.