Zeros of the derivative of a p-adic meromorphic function

Let K be a complete algebraically closed field of characteristic 0 and let f be a transcendental meromorphic function in K. A conjecture suggests that f′ takes every values infinitely many times, what was proved when f has finitely many multiple poles. Here we can generalize the conclusion just by assuming that there exist positive constants c, d such that number of multiple poles inside the disk |x|⩽r is less than crd for all r⩾1. Applications are given to entire functions g in K such that g′ divides g, to links between residues and zeros of functions admitting primitives and finally to the p-adic Hayman conjecture in the cases that are not yet solved.