Martin's initial one-month stay at Pride Park expired after last Saturday's 2-1 home victory over Leicester, in which the 24-year-old scored his first goal for the Championship club.
A deal is now in place with Norwich to keep Martin in Derbyshire until May, with the paperwork due to be completed next week.
"We're happy Chris is sticking around and we were delighted to see him score against Leicester last weekend," Rams boss Nigel Clough told the club's official website.
"We have been talking to Norwich for around a week now, so it's good to get something sorted.
"We've been very encouraged by what we've seen from Chris and feel there's more to come from him too.
"He's not played too many games this season so he's been a little short in terms of match fitness and that side of his game will improve with the more games he plays, but in terms of his quality and use of the ball we have been very pleased with him."