All of the hull panels in First Mate, including the narrow flat bottom panel, are made from 6mm/1/4" marine plywood. Wayne Jorgensen, who helped build the first boat from the design to hit the water, expressed some concern about the stiffness of such a thin bottom panel. I wasn't particularly concerned, as the panel is only 856mm/34" wide at the widest point, and is very well supported by the centreboard case, 6 bulkheads and frames, and most importantly, by a 66mm x 19mm (2-9/16" x 3/4") external keel batten firmly glued to the bottom of the boat. In addition, I sheathed the boat I'm building with 200gsm/6oz glass cloth in epoxy.
In a beach-cruiser, light weight is of critical importance for a solo sailor, and I also wanted to economise on plywood, getting the most possible out of standard sheets - it made sense to use a standard thickness for all panels. Having said that, there is nothing to prevent you from using 9mm/3/8" ply for the bottom panel if you wish.
As part of the design, I drew an adjustable set of foot braces to make rowing more efficient, and also to give something against which to lock your heel to prevent sliding across the boat when heeling under sail. These foot braces gave me yet another opportunity to stiffen the bottom panel. Following are a few images of the set I've just installed in the First Mate which I am (slowly) building: -
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