Lambert, 19, scored five goals in eight appearances for the Robins last season after signing from Norwich City.
Manager Martin Carruthers said: “I’m delighted to retain the services of Kole Lambert.
“There is so much more to come from him and once the season gets started I’m sure we will see a different player. Kole had just started to find some form and confidence leading up to the lockdown, but we know we can unlock that potential and get the fans buzzing with his goals and finesse.
“A huge thank you to Dave, Mark & Andrew who have backed myself and Wayne in putting this fantastic squad together. We are almost there and there maybe one or two additions to get us ready to hit the ground running.”
Lambert joined Norwich from Basford United in January 2019 following a successful trial.
He had previously been part of the academy set ups at Notts County and Nottingham Forest.