Katla is a threatening volcano in Iceland, partly covered by the M\'yrdalsj\"okull ice cap. The volcano has a large caldera with several active geothermal areas. A peculiar cluster of long-period seismic events started on Katla's south flank in July 2011, during an unrest episode in the caldera that culminated in a glacier outburst. The seismic events were tightly clustered at shallow depth in the Gvendarfell area, 4 km south of the caldera, under a small glacier stream on the southern margin of M\'yrdalsj\"okull. No seismic events were known to have occurred in this area before. The most striking feature of this seismic cluster is its temporal pattern, characterized by regular intervals between repeating seismic events, modulated by a seasonal variation. Remarkable is also the stability of both the time and waveform features over a long time period, around 3.5 years. No comparable examples have been found in the literature. Both volcanic and glacial processes can produce similar waveforms and therefore have to be considered as potential seismic sources. Discerning between these two causes is critical for monitoring glacier-clad volcanoes and has been controversial at Katla. For this new seismic cluster on the south flank we regard volcano-related processes as more likely than glacial ones for the following reasons: 1) the seismic activity started during an unrest episode involving sudden melting of the glacier and a j\"okulhlaup; 2) the glacier stream is small and stagnant; 3) the seismicity remains regular and stable for years; 4) there is no apparent correlation with short-term weather changes, such as rain storms. We suggest that a small, shallow hydrothermal system was activated on Katla's south flank in 2011, either by a minor magmatic injection or by changes of permeability in a local crack system.