A lot depends on the tools at hand and your experience. With a properly set up band saw, simple resawing jig and a good blade, it can be done to any width you wish, some waste will go into the kerf.

You can cut them by hand with a fine/thin pull saw easily, with very little waste, but does take some care. Mark the cut line on all 4 sides and cut a kerf on each side, then cut in half using the kerfs as a guide. Horn cuts easily and quickly once cut to width, about 1 in or so.

With any resawing, cutting thin strips can easily produce potato chips, depending on the tension in the material, though they can be straightened. Natural materials are… Natural.

There are some good recent threads on buying Horn from Ebay sellers. You may want to start with the thinner material and experiment with resawing the thicker stuff, there is a learning curve…