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If you can't find one for sale then you could probably just rig up your own. That container just looks like half a contact lens case (you can get ones with pop-off lids rather than screw-off lids) and you could just bung some vaseline in there. How crafty are you?
Years ago I was given a promotional lip balm at a race. It was worn on your watch strap. It is the most useful thing I have ever owned. It ran out quickly, but I have been refilling it for years. However, the inevitable has happened and it has been lost
Does anyone know where I might get another? It was this product (except marked with Asics logo) but none of the links here work. I have searched endlessly on-line and have been able to find nothing else like it.