We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) noncommutative p-extension F-infinity of a totally real number field F such that the finite part of its Galois group G is a p-group of exponent p. We first calculate the Whitehead groups of the Iwasawa algebra Lambda(G) and its canonical Ore localization Lambda(G)(S) by using Oliver and Taylor's theory of integral logarithms. This calculation reduces the existence of the noncommutative p-adic zeta function to certain congruences between abelian p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne and Ribet's theory and a certain inductive technique. As an application we prove a special case of (the p-part of) the noncommutative equivariant Tamagawa number conjecture for critical Tate motives.