Looks like it might be a version of the coat of arms for the city of Paris (since the 14th Century). There are numerous versions of it, with different shield shapes having ships with between one and three masts, with or without the castellate crown above it, and with or without the oak and laurel leaves around it.
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In certain periods, the fleurs-de-lis were replaced by bees, and then by stars, before the old coat of arms was restored in 1853. The three medallic badges (as in the third example above) were then sequentially added to the full armorial in 1900, 1919 and 1945.
Your button looks like it might have a motto above. If you can see any part of ‘FLUCTUAT NEC MERGITUR’ as the letters then it’s definitely for Paris, since that’s the city’s motto, meaning “[She] is tossed [by the waves], but does not sink”.
I have a feeling that, if this is French, it’s not military but civic authority of some kind. This one is similar to yours and was worn by Parisian police officers and prison guards:
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Are there any backmarks which might help?