## It seems so, yes.
After the game played between Death and Life-in-Death, in which the latter wins the soul of the Ancient Mariner while the former claims all his crewmates, the Mariner is left all alone upon the ship. He is driven to distraction by the corpses, but says he **cannot die** :
> Fear not, fear not, thou Wedding-Guest!
> This body dropt not down.
>
> [...]
>
> And a thousand thousand slimy things
> Lived on; and so did I.
>
> [...]
>
> Seven days, seven nights, I saw that curse,
> And yet I could not die.
>
> \-- Part IV
It's also worth noting that he goes **at least a week without water** ("Water, water, every where, / Nor any drop to drink" comes earlier, in Part II, and he only manages to drink at the start of Part V). This too suggests that he's supernaturally unable to die.