According to the scenario, the Kobayashi Maru was located in Gamma Hydra, section ten, nineteen periods out of Altair VI. The vessel had struck a gravitic mine, and its hull was breached. The ship had lost all power, life support was failing, and had sustained many casualties. (Star Trek II: The Wrath of Khan)
James T. Kirk said "the Kobayashi Maru has set sail" on his way to hijack the USS Enterprise from Earth Spacedock right after breaking Dr. Leonard McCoy out of jail. (Star Trek III: The Search for Spock)
The term "Kobayashi Maru" was used by McCoy when imprisoned in the Rura Penthe dilithium mines in 2293. He commented to James T. Kirk in their first night there, "One day, one night... Kobayashi Maru," making a throat-slash motion before Kobayashi Maru, to suggest they weren't going to last another day. (Star Trek VI: The Undiscovered Country)
The Pocket ENT novel Kobayashi Maru, set in 2155, details the historical events which became the Academy simulation. Meanwhile, David Mack's novel A Time to Heal, Ensign Carmona identifies the Kobayashi Maru as having been lost in the Tezel-Oroko system, the location of the planet Tezwa, where most of this book occurs.
The novel The Kobayashi Maru, written by Julia Ecklar, details the Kobayashi Maru tests taken by Kirk, Chekov, Scotty, and Sulu during their respective enrolments at Starfleet Academy. The depiction of the Kobayashi Maru on the cover of the novel was that of a Tritium-class starship (β), originally designed by Rick Sternbach for the Spaceflight Chronology. (p. 130) Peter Kirk became the second Kirk to beat the no-win scenario in A.C. Crispin's Sarek, although doing so in a manner more in keeping with the rules.
An updated version of the test was featured in Peter David's New Frontier book Stone and Anvil. In the book Elizabeth Shelby, then a cadet, was given the task of updating the scenario, with one of its first participants being Mackenzie Calhoun (also a cadet). In the updated version of the scenario the Klingons were replaced by Romulans, and an extra degree of difficulty was added with a radiation leak on the freighter. Calhoun's solution to it was to fire upon and destroy the freighter on the grounds that the Romulans were using it as bait to provoke an incident and possibly a war.
In Japanese "kobayashi maru" can be translated as "little wooden boat", an apt metaphor for its defenceless nature. In The Q Gambit, Part 1, Nyota Uhura translates the word "Kobayashi" as "Smallwood", the name of the Federation civilian ship they encounter being attacked by Klingons (in reality, a test put forward by Q).
Star Trek Online, which features a version of the no-win scenario, shows the Kobayashi Maru as being a variation of the Miranda-class, with a number of small hangar bays around the bridge module, the nacelles mounted on the upper part of the ship (with the lower pylons occupied by cargo containers), as well as an extended flight deck in the aft section (similar to the Soyuz-class).
The VR game Star Trek: Bridge Crew features a mission where the players' ship encounters the Kobayashi Maru under attack from Klingon birds-of-prey, but in an unusual break from tradition, it is entirely possible for the players to rescue at least some of the crew members while fending off the Klingons, and then escape with their own ship intact.