Let K be a complete algebraically closed p-adic field of characteristic zero. We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove results of uniqueness in value sharing prob-lems, both on K and on C. Let P be a polynomial of uniqueness for meromorphic functions in K or C or in an open disk. Let f , g be two transcendental meromorphic functions in the whole field K or in C or meromorphic functions in an open disk of K that are not quotients of bounded analytic functions. We show that if f′P′( f ) and g′P′(g) share a small function α counting multiplicity, then f=g, provided that the multiplicity order of zeros of P′satisfy certain inequalities. A breakthrough in this pa-per consists of replacing inequalities n≥k+2 or n≥k+3 used in previous papers by Hypothesis (G). In the p-adic context, another consists of giving a lower bound for a sum of q counting functions of zeros with (q-1) times the characteristic function of the considered meromorphic function.