0.5 inch ( 12 mm ) Starboard should be available at that "marine store on every cormner", or from Piedmont Plastics ( or other plastics supplier ). You might call King Starboard to get the vender most appropriate for you.
Note that Starboard, which might bve available in a good color for this project, is HDPE, ans as such has a few characteristics you should know about. In spite of this it is often being used in what I might call a structural usage, the material is not designed for structural usage. It also has a very high coefficient of thermal expansion, which means that a) it expands and contracts at a rate different from fiberglass, and b) if the sun shines on it unevenly, it can warp ( at least, temporarily ). But it might be a good product for this. Note that it is virtually impossible to glue anything to it, or paint it. Graphite ( ie: pencil marks ) are essentially permanent.
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I would use exterior plywood, "paint" it with an undercoat of epoxy, sealing the panel as well as edges, then paint and/or carpet as desired. Mount it or re-design it so that it does not stand in water. Done as I describe, it will be fine. It will be "casually waterproof" and not suffer from the periodic washdown, as long as it does dry out after a few hours, and is normally dry. My design will not last long term, standing in water continuously...
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The panel you describe is almost certainly plywood ( maybe exterior, maybe not ) with one side coated in Melamine. This is a moderately common practice. It is fine as long as immersions are brief, and the panel is normally dry. It will fail ( as occurred ) when the panel does not dry out after these brief immersions.
Most of the plywood in boats ( at least those of recent manufacture ) use "interior grade" or equivalent for anything above the waterline, or not constantly in water contact. It saves weight, and $$$. Personally, I do not care for the practice, but as far as I can tell, the practice is nearly universal in all but "premium" grade boats ( and even then such materials are used in "dry spaces".
My preference is for all materials and structures on a boat to be at least "water resistant/tolerant". But that does cost more and usually also weighs more.