Let p be a prime number and x an irrational p-adic number. The multiplicative irrationality exponent m(x) (resp. uniform multiplicative irrationality exponent u(x)) of x is the supremum of the real numbers m for which the inequalities 0 < |a b|^{1/2} < A and |b x - a|_p < A^{-m} have a solution in integers a, b for arbitrarily large real numbers A (resp., for every sufficiently large real number A). We show that these exponents of approximation can be expressed in terms of exponents of approximation attached to a sequence of rational numbers defined in terms of the Hensel expansion of x. We discuss the set of values taken by the exponents m and u at irrational p-adic numbers and compute the values of m and u at the Thue-Morse p-adic number.