Let K be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value. We show that if the Wronskian of two entire functions in K is a polynomial, then both functions are polynomials. As a consequence, if a meromorphic function f on all K is transcendental and has finitely many multiple poles, then f takes all values in K infinitely many times. We then study applications to a meromorphic function f such that f + bf 2 has finitely many zeroes, a problem linked to the Hayman conjecture on a p-adic field.