Are you asking about sending mechanically scanned texts, or texts laboriously entered by hand?

Part One: Sending a book by ordinary telegraph.

If you mean sending a book via ordinary telegraph with someone constantly hitting the keys in Morse code, I think there are examples of persons sending very long messages by telegraph.

Clement Lounsberry, editor of the Bismarck Tribune in North Dakota, sent Mark Kellogg as a reporter with the Dakota Column in 1876. Kellogg was killed at the Little Big Horn 25 June 1876:

When Clement Lounsberry learned of the defeat of Custer's force and Kellogg's death, he "worked tirelessly throughout the night" to produce a special edition of The Bismarck Tribune. Published on July 6, 1876, the article was the battle's first full account. Lounsberry also telegraphed the news, including Kellogg's correspondence, to a number of eastern newspapers, including the New York Herald. Two letters written by Kellogg were published posthumously by the Herald on July 11, 1876.

And I think I remember that it took a long time to transmit the entire account of the battle to the New York Herald, and once the story had been transmitted Lounsberry had the telegraph line tied up for hours by transmitting any nonsense he could think of over the telegraph, thus preventing anyone from sending the story to competing newspapers.

If that story is correct, and I don't know whether it is, it would be an example of someone transmitting tens of thousands of words by telegraph at a single session, which thus would be the equivalent in length of transmitting a full length book by telegraph.

I note that books published in the UK were sometimes also published in the USA, and vice versa. The first transatlantic telegraph cable was finished in 1858 but only worked for three weeks. The next transatlantic telegraph cables were installed in 1865 and 1866 and lasted much longer.

Presumably book manuscripts would normally be mailed across the Atlantic, taking a week or two by ship, but if publishers were in a hurry they might theoretically have had used the transatlantic cables to transmit the manuscript across the Atlantic. But I don't know when — or if — that was first done in the 19th century.

Part Two: 19th century faxing of books?

I have heard of Fax machines sending facsimiles of writing and pictures over wires.

Once when rereading Frederick Pohl's science fiction story "The Children of Night" I noticed a mention of Fax machines. The fictional date of the story was sometime in the early 20th century, probably before 2020 — I don't have access to the paper where I once tried to calculate the fictional date.

But the story was first published in 1964.

And I wondered whether fax machines were already used by 1964 or whether Pohl was writing about a future technology in 1964. So I researched the history of fax machines and found they were used by 1964.

Wikipedia: Fax#Wire_transmission:

Scottish inventor Alexander Bain worked on chemical mechanical fax type devices and in 1846 was able to reproduce graphic signs in laboratory experiments. He received British patent 9745 on May 27, 1843 for his "Electric Printing Telegraph".

Frederick Bakewell made several improvements on Bain's design and demonstrated a telefax machine. The Pantelegraph was invented by the Italian physicist Giovanni Caselli. He introduced the first commercial telefax service between Paris and Lyon in 1865, some 11 years before the invention of the telephone.

In 1880, English inventor Shelford Bidwell constructed the scanning phototelegraph that was the first telefax machine to scan any two-dimensional original, not requiring manual plotting or drawing. Around 1900, German physicist Arthur Korn invented the Bildtelegraph, widespread in continental Europe especially following a widely noticed transmission of a wanted-person photograph from Paris to London in 1908, used until the wider distribution of the radiofax. Its main competitors were the Bélinographe by Édouard Belin first, then since the 1930s the Hellschreiber, invented in 1929 by German inventor Rudolf Hell, a pioneer in mechanical image scanning and transmission.^{[citation needed]}

The 1888 invention of the telautograph by Elisha Gray marked a further development in fax technology, allowing users to send signatures over long distances, thus allowing the verification of identification or ownership over long distances.

On May 19, 1924, scientists of the AT&T Corporation "by a new process of transmitting pictures by electricity" sent 15 photographs by telephone from Cleveland to New York City, such photos being suitable for newspaper reproduction. Previously, photographs had been sent over the radio using this process.

The Western Union "Deskfax" fax machine, announced in 1948, was a compact machine that fit comfortably on a desktop, using special spark printer paper.

So in 1924 fax machines using telephone wires were able to send newspaper quality photographs. Clearly that system could theoretically have been used to send images of the pages of a full length book. As could the desktop fax machines introduced in 1948.

From the description of the Telautograph, I think that someone would have to write at the sending station for the writing to be transmitted, and it was mostly used for transmitting signatures.

Wikipedia: Telautograph

The German Bildtelegraph in the early 20th century scanned and transmitted photographs. I guess it could have been used to laboriously send the pages of a full length book.

The pantelegraph used a regulating clock with a pendulum which made and broke the current for magnetizing its regulators, and ensured that the transmitter's scanning stylus and the receiver's writing stylus remained in step. To provide a time base, a large pendulum was used weighing 8 kg (18 lb), mounted on a frame 2 m (6 ft 7 in) high. Two messages were written with insulating ink on two fixed metal plates; one plate was scanned as the pendulum moved to the right and the other as the pendulum moved to the left, so that two messages could be transmitted per cycle. The receiving apparatus reproduced the transmitted image by means of paper impregnated with potassium ferricyanide, which darkened when an electric current passed through it from the synchronized stylus. In operation the pantelegraph was relatively slow; a sheet of paper 111 mm × 27 mm, with about 25 handwritten words, took 108 seconds to transmit.

The most common use of the pantelegraph was for signature verification in banking transactions.

Wikipedia: Pantelegraph

From that description I supposed it was theoretically possible to transmit a full length book on a pantelegraph line in the 1860s, which would have been very time consuming and expensive — presumably someone could have carried it on the train much faster, or even walked must faster than it would have taken to transmit an entire book.