We prove that for any prime p and any integer k > 2,there exist in the ring Zp of p-adic integers arbitrarily long sequences whose sequence of k-th powers 1) has its k-th difference sequence equal to the constant sequence (k!); and 2) is not a sequence of consecutive k-th powers. This shows that the analogue of Buchi's problem for higher powers has a negative answer over Zp.This result for k = 2 was recently obtained by J. Browkin.