I received a helpful suggestion to use the book Dakota Imprints to determine if The People’s Problem and Its Solution by William H. Lyon was the first book published in the Dakota Territory. It does not list Lyon’s book, but the first book it does include was published in the same year as Lyon’s, 1886.
Dakota Imprints lists items earlier than 1886, but they all appear to be Dakota Territory publications, broadsides, and an industry circular. The book that was published in the same year as Lyon’s was History and Resources of Dakota, Montana, and Idaho by Moses K. Armstrong.
Dakota Imprints and Armstrong’s book are in the Chilson Collection. Lyon’s book is in the William H. Lyon Papers.