In 1931, the German publisher Albatross Books was the first to produce a mass-market paperback book, but the approaching WWII prevented further marketing. In 1935, the British publisher Penguin Books began the paperback revolution in the English language book-market. In 1939, the Pocket Books label appeared in the United States. Thereafter, the term “pocket book” was synonymous with English language paperbacks in North America.
During WWII, American publishers collaborated to produce 1180 pocket-sized Armed Services Editions of popular books that were distributed free of charge to 16 million service men and women .
Read a fascinating article by Louis Menand: The Birth of Pulp Fiction – The New Yorker.
Bantam Books is an American publishing house owned entirely by parent company Random House, a subsidiary of Penguin Random House; it is an imprint of the Random House Publishing Group. It was formed in 1945 by Walter B. Pitkin, Jr., Sidney B. Kramer, and Ian and Betty Ballantine. It has since been purchased several times by companies including National General, Carl Lindner’s American Financial and, most recently, Bertelsmann; it became part of Random House in 1998, when Bertelsmann purchased it and merged it with Bantam Doubleday Dell.[1] It began as a mass market publisher, mostly of reprints of hardcover books, with some original paperbacks as well. It expanded into both trade paperback and hardcover books, including original works, often reprinted in house as mass-market editions. Wikipedia
The trio that founded Bantam did so to publish the Instruction Manuals that were tucked in the pockets of the aircraft leaving the US & England to fight overseas. They were all softcover and published by Penguin in the UK. I don’t think Penguin had a US Publishing presence until after the end of WWII.