During the Civil War, the Episcopal Church in the Confederate States was temporarily separated from the rest of the Episcopal Church. There were three complete printings of a Confederate Prayer Book, all done in London, and none apparently formallly authorized. They had to be smuggled into the South, and only one of the printings actually made it through the blockade, the other two being captured and mostly destroyed. I have never seen a copy of a Confederate BCP, but my understanding is that they were completely identical to the 1789 BCP, with the exception of the necessary changes in prayers for the President and Congress. Additionally, a number of partial Books of Common Prayer were printed locally.

The University of North Carolina has placed online a number of texts connected with the Episcopal Church in the Confederate States. Those of interest here include:

• An abridgement of the Book of Common Prayer, The Order for Daily Morning and Evening Prayer, According to the Use of the Protestant Episcopal Church in the Confederate States of America, Together with the Ante-Communion Office and a Selection of Occasional Prayers from Various Offices of the Book of Common Prayer. This book, printed in Atlanta, was intended for use by soldiers, and contains most of the BCP services which do not require a priest - e. g., it omits Holy Communion, Baptism, Marriage, etc. In addition, while the Book is officially authorized, the services it does have are nevertheless a bit different — for example, in Morning and Evening Prayer, some of the introductory sentences and the Nicene Creed are omitted, and the selection of Psalms and Hymns is different. It is also available from the Internet Archive.
• A Prayer Book for the Camp, issued by the Diocese of Virginia and also intended for use by the Military. This book includes Morning Prayer, modified along the lines mentioned above, the Litany, ante-Communion, the Burial service, the Collects, several prayers, and hymns. Also available as PDF graphics from the Internet Archive.
• A Catechism, identical to that in the 1789 BCP.
• A Catechism to be Taught Orally to those who cannot Read, intended for the instruction of slaves.
• A Calendar of the Days and Daily Lessons of the Year 1862. The daily lectionary; also includes a clergy directory.
• A number of other items not connected with the Book of Common Prayer, such as Convention Journals.

The Internet Archive also has a partial BCP, containing a combined Morning and Evening Prayer, the Litany, Prayers & Thanksgivings, some additional prayers, and a few hymns, printed in Charlottesville.

Also, Project Canterbury has online a History of the [Episcopal] Church in the Confederate States, which, among many other things, describes the various Confederate Prayer Books in detail.

At the end of the Civil War, Dioceses in the South again became part of the Episcopal Church, and resumed using the 1789 Book of Common Prayer.