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

A divergent horocycle in the horofunction compactification of the Teichm\"uller metric