Let K be a complete algebraically closed p-adic field of characteristic zero. Let f, g be two transcendental meromorphic functions in the whole field K or meromorphic functions in an open disk that are not quotients of bounded analytic functions. Let P be a polynomial of uniqueness for meromorphic functions in K or in an open disk and let α be a small meromorphic function with regards to f and g. Here we present the following results: if f ′ P ′ (f) and g ′ P ′ (g) share α counting multiplicity, then we show that f = g provided that the multiplicity order of zeros of P ′ satisfy certain inequalities. If α is a Moebius function or a non-zero constant, we can obtain more general results on P. Further, when f, g are entire analytic functions or analytic functions inside an open disk, we can obtain a new result improving that published by the third author