Yes, they are, according to the novelization, and this is how it is in the film.
From the Fantastic Four: Rise of the Silver Surfer movie novelization, page 192: It's pointed out when the Silver Surfer is questioned by Susan Storm a.k.a Invisible Woman about the "source" of his power and how it's the "key" to Earth's destruction by drawing his master Galactus, who bears the moniker "the Destroyer" in the novelization:
"Could you stop the Destroyer from coming if you wanted to?" she asked over her shoulder, keeping her gaze on the miles of artic land around them.
The Surfer answered her quickly, like a child who longed to be awarded. "It's not I who draws the the Destroyer (Galactus) here. It's the beacon."
"What beacon?" she asked.
"The source of my power," he said.
"The board?" she thought to herself. The earth is doomed to a fiery apocalyptic demise because of a surfboard?
"Take joy in the last few hours you have left," the Surfer added. "For it is nearly here."
In the comics, on the other hand, the Silver Surfer's power comes from the Power Cosmic
, not his surf board. On more than one occasion the board has been destroyed and he has simply remade it. In the time since 2007, this had been seen in the Annihilation
mini-series. The Silver Surfer has complete mental control over his board, regardless of whether or not he is in physical contact with it. While the aforementioned is true, it also true that (at least originally) his surf board contained his weak points in the comics. While his powers did not come from the board, damaging the board would damage the Surfer.