Non-liftability of the field of norms of the p-adic false Tate curve extension

Let p≥3 be a prime number. We prove that the action of the Galois group on the field of norms of the p-adic false Tate curve extension Qₚ(ζp^∞,p1/p^∞) of Qₚ is not liftable to its Cohen ring with respect to any admissible coefficient field.