No. the conjunctiva is the thin, moveable, and semi-transparent layer that covers the white part of the eye (the sclera). It does not extend past the edge of the cornea (the limbus). The iris sits a little behind the cornea. The conjunctiva only extends the other way back about half way, that is, to about the equator of the eyeball where it doubles back forward, covering the inside of the eyelids.
Many contact lens patients worry that a contact might be able to slip all the way around to the very back of the eyeball, or even into the eye socket itself. But this does cannot happen.