  The Thue--Morse sequence is a prototypical automatic sequence found in
diverse areas of mathematics, and in computer science. We study occurrences of
factors $w$ within this sequence, more precisely the sequence of gaps between
consecutive occurrences. This sequence is substitutive; we prove that it is not
automatic as soon as the length of $w$ is greater than $1$, thereby answering a
question by J.~Shallit in the affirmative.
