If your Flic button doesn't connect to the app or doesn't light up anymore, it's most likely that the battery has run out.
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To change the battery of a Flic button* you can hold the bottom of the Flic with one hand and the top with another hand, and twist in opposite directions.

You can also stick the Flic to a flat surface and twist the top off.
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Place a new CR2016 battery with the positive side facing the back piece (the side with the adhesive). Press hard and twist the two pieces together again, until they click shut.

The following video shows the full process of replacing the battery:

If your Flic button shines a constant red light, the battery has been moved or placed incorrectly. Remove the battery and place it in the button again.

Also note that the battery must be new, not more 6 months old, because natural discharges might be produced and the voltage level might be not high enough to boost the chip in the button.
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* Note that Flic Single (Lights, Location, Music, Find and Selfie) cannot be opened and therefore their battery cannot be changed.