The 20-year-old versatile attacker will be with Town until at least next summer – with the option of a further year beyond that as well.
And Ricketts hopes to see Barnett, who was on loan at Shropshire neighbours AFC Telford United this season, become a regular part of his Town squad in 2020/21.
"I'm really pleased that Barney has signed his new deal," said Ricketts "As a club, we really like to try and promote the youth coming through and we want to get as many local lads as we can coming through the academy and into the first team.
"Barney has made great strides since signing his first deal with the club.
"He's been out on loan a couple of times and I've seen him develop in that time so I'm now looking forward to seeing him push on next season."
Barnett has been with the club 12 years and signed his first professional contract with Town in 2017 at the age of 17 after rising through the academy ranks.
He tweeted last night: "12 years. Buzzing to sign a new deal and get back to playing!"
And it has also been announced that 20-year-old goalkeeper Cameron Gregory had signed a new two-year deal with the option of an extra 12 months.
Ricketts added: "I'm really delighted that Cam is staying at the club.
"Similar to Ryan Barnett who also signed today, Cam is a young player who has come through the academy. He's been on loan this year and done really well so we're really keen to keep him and watch how he progresses and develops."