  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 $\alpha$ be a small meromorphic
function with regards to f and g. If f'P'(f) and g'P'(g) share $\alpha$
counting multiplicity, then we show that f=g provided that the multiplicity
order of zeroes of P' satisfy certain inequalities. If $\alpha$ is a Moebius
function or a non-zero constant, we can obtain more general results on P.
