No, it's not possible. The other galaxies we see are to radically different. Additionally, if we are the surface volume of hyperspace, then the universe should be closed. Our best estimates and observations indicate it's flat. Let me address both of these in more detail.
As for the other galaxies. First of all, there's the Andromeda galaxy. That is a galaxy that is bigger than ours (has more mass) and is very close. In fact, we know the Andromeda galaxy will collide with the Milky Way at some point in the future. To continue off all of this, when we look out, we can see clusters of galaxies where the angular separation between two galaxies is minimal and the physical distance we measure is relatively small, yet they appear radically different. On top of that, there are galactic mergers and collisions of galaxies that we can observe. If all are just images of the Milky Way, we'd never see two galaxies collide, yet things like the Mice galaxies show that they do.
Let's also consider that the Milky Way has its own satellite galaxies that are very clearly not images of the Milky Way.
But on to the issue of spatial curvature. If, say, the universe was a hypersphere and we were in the surface volume, the universe would be considered closed (a curvature term). In the case of such curvature, some things would be evident that clearly aren't. Consider simply geometry. On a flat piece of paper, draw a triangle. The angles inside it always sum to $180^\circ$. Now consider a sphere. Draw a triangle on the surface. Any triangle that covers a significant portion of the surface does not have angles that sum to $180^\circ$. The same would be true for the surface of a hypersphere. And since you propose the light would travel all the way around, that is significant enough for these effects to be measurable.
Now, it's possible that we are on the surface of a hyperspace, however it would have to be large enough that the curvature within our visible portion of the universe is nearly zero. That means it would be too large for light to travel all the way around and come back to us.
Furthermore, if the other galaxies we see are just images of the Milky Way, why don't we see more of them? Our galaxy radiates light in all directions. It stands to reason that if some light is bent back towards us, this would be a continuous phenomenon. So we should see light from our galaxy coming back from all directions, not just specific enough ones to make an image of the Milky Way.
Believe it or not, I could go on, but I think you get the idea that the other galaxies we see are definitely other galaxies.