  We give an example of a horocycle in the Teichm\"uller space of the
five-times-punctured sphere that does not converge in the Gardiner--Masur
compactification, or equivalently in the horofunction compactification of the
Teichm\"uller metric. As an intermediate step, we exhibit a simple closed curve
whose extremal length is periodic but not constant along the horocycle. The
example lifts to any Teichm\"uller space of complex dimension greater than one
via covering constructions.
