  We show that the Kakimizu complex of a knot may be locally infinite,
answering a question of Przytycki--Schultens. We then prove that if a link $L$
only has connected Seifert surfaces and has a locally infinite Kakimizu complex
then $L$ is a satellite of either a torus knot, a cable knot or a connected
sum, with winding number 0.
