The claims potentially made by this seeker of life-prolonging remedy (9)
I believe the answer is:
alchemist
'this seeker of life-prolonging remedy' is the definition.
(alchemists sought the elixir of life)
'the claims potentially' is the wordplay.
'potentially' is an anagram indicator (letters have the potential to spell a new word).
'the'+'claims'='theclaims'
'theclaims' is an anagram of 'ALCHEMIST'.
'made by' is the link.
(Other definitions for alchemist that I've seen before include "Seeker after gold" , "The St Michael who tried to turn metal to gold" , "St Michael in role of one turning base metal into gold" , "St Michael (anag)" , "such as Paracelsus" .)