Liam has made 12 appearances for the Glovers and has been a standout signing by manager Darren Way.
18-year-old Walsh has put in numerous man-of-the-match performances since he signed early in January.
Darren Way said: "Liam was my first signing as manager and he has shouldered the responsibility of helping this club away from the position we were in like a man. I am looking forward to seeing where Liam's career goes in the future. I am sure he has a very, very bright future ahead of him.
"We have gained a valuable trust and relationship with Everton and I look forward to working with them again in the future"