p-Adic meromorphic functions f′P′(f), g′P′(g) sharing a small function

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. If f′P′(f) and g′P′(g) share α counting multiplicity, then we show that f=g provided that the multiplicity order of zeroes of P′ satisfy certain inequalities. If α is a Moebius function or a non-zero constant, we can obtain more general results on P.