An algebra B is called bicommutative if it satisfies the identities

x(yz) = y(xz) (left commutativity)

(xy)z = (xz)y (right commutativity)

for all x,y,z in B. Wild automorphisms are constructed in two-generated and three-generated free bicommutative algebras. Moreover, for any n \geq 2, a wild automorphism is constructed in the n-generated free associative bicommutative algebra which is not stably tame and cannot be lifted to an automorphism of the n-generated free bicommutative algebra. We also offer fast algorithms for calculating the Gelfand-Kirillov dimension of finitely presented bicommutative algebras.