Galloway has been called into the Toffees' first-team this season to cover for Leighton Baines, and he has done a brilliant job.
The 19-year-old has coped with every challenge thrown at him, and Martinez is excited for the academy graduate's future on Merseyside.
"Brendan keeps surprising everyone," the Spanish boss said. "His maturity is very impressive. He gets ready from game to game, and I haven't seen him needing time to rest. You need to measure that week to week, and let football make the decision.
"I don't think there is a general role that if someone is 19 and he hasn't played before, that's the amount of games you need to give him. We have been very pleased in the manner he has adapted to the different demands, and by the way he has continued to grow. Clearly, we have someone special for the future."
With Baines nearing full fitness now, Galloway's time in Everton's first-team is likely to be limited, but he has done his chances for the future no harm.
Baines remains one of the finest left-backs in the Premier League, but Galloway has shown that he is an able replacement, if needed.
The powerful full-back has been tough for opposition players to beat, while he has also got forward with aplomb.
Galloway's next match could come at the weekend, when Everton take on Arsenal, in what looks like another tough test for the young defender.