[spoiler alert]
By Xenocide, they solved the FTL travel problem, thanks to Jane
Piers Anthony’s five-volume Bio of a Space Tyrant has nary a warp gate, just a lot of goofy politics and explicit sex.
I t hink that you’re forgetting Heinlein’s book Universe. I really don’t want to go into more detail, as most of what I could say would be spoilers, but again, Heinlein does the First Story Of Its Kind In SF.
I’m tired of making my usual rant about the term sci-fi, so let’s just all consider it DONE, all right?
The Sparrow and Children of God, neither contain FTL travel or communications.
Technically those particular stories fall into a gray area, as they are all set in his Known Space future history (except for ‘Inconstant Moon’, which doesn’t even involve space travel and is set in modern times). In his Known Space stories FTL is possible but was not known to humans until after we had already colonized a few worlds with STL colony ships.
All of his stories set in his ‘The State’ future history (‘A World out of Time’, ‘Night on Mispek Moor’, ‘The Integral Trees’, and ‘The Smoke Ring’) are completely without FTL, though, as are ‘Destiny Road’, ‘Legacy of Heorot’, and ‘Beowulf’s Children’.
Poul Anderson’s The Boat of a Million Years
Offhand, I don’t believe any of Spider Robinson’s works, and there are a ton of them, have any FTL-based stories. He does rather play around with time travel, though. And there are reams of stories that go forward and back rather than out for their excitement.
Yersinia Pestis, I remember the Zelazny story you’re thinking about - I don’t remember the name, but it’s in the short story collection The Doors of his Face, the Lamps of his Mouth.
Recently, Alastair Reynolds’ novels (Revelation Space and Chasm City) are set in a future universe with no FTL (interstellar ships, because they travel at relativistic speeds, are referred to as “lighthuggers”). I think the same is true for Linda Nagata’s Deception Well and Vast.
(Ursula Le Guin has slower-than-light interstellar ships, but they have FTL communications, so I guess they don’t count.)
A science fiction novel that doesn’t have the level of science we have now, let alone FTL travel: 1984.
pan
A lot of Clarke’s stuff has no FTL, but offhand, the only one I can think of by name is Imperial Earth.
As of the first Rama book, there was no indication that Rama could go FTL. If only he’d stopped there . . .
Likewise, cyberpunk usually don’t have FTL, and frequently doesn’t have space travel. Walter Jon Williams’ Hardwired never goes beyond LEO, Days of Atonement never even got that far. While Voice of the Whirlwind has a little FTL (which is deliberately left kind of vague) in the backstory, the story itself stays inside of the solar system.
I don’t think George Alec Effinger’s Marid Audron novels ever went higher than a scramjet.
Movies, hmm. A bunch of late-80s, early 90s action-SF movies didn’t have FTL, and usually skipped space flight, as well. Terminator, Fortress, (Not that I recommend WATCHING it, of course) The Running Man, Total Recall, like that.
Near-future is fairly common in manga and anime, and often doesn’t have FTL. Ghost in the Shell, Venus Wars, Akira, Patlabor, and Appleseed, (avoid the anime. The manga is terrific, however) just off the top of my head, and only counting movies.
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“Somehow, the answers to all these questions is: White guys in jumpsuits.”
Total Recall had FTL communication, people were having normal conversations with people on Mars, and in the absolutely horrid novelization by Piers Anthony he mentions the technology that allows this.
Did the Rama sequels have FTL?
As for other Clarke novels with slower-than-light interstellar travel, The Songs of Distant Earth has already been mentioned. I think the aliens in Childhood’s End had slower-than-light starships too - one of the characters stow away in one and come back hundreds of years later. Fountains of Paradise had a slower-than-light alien probe that made contact with earth.
Most stories that deals with interplanetary flights (not interstellar) obey the laws of relativity. Clarke has written a few, like Earthlight and A Fall of Moondust as well as Imperial Earth, like you said, but so have many other authors. Heart of the Comet by Gregory Benford and David Brin deserves a mention though - it dealt with long-distance long-term space travel (a 76-year journey on Halley’s Comet).
Speaking of Gregory Benford, did his Galactic Center series (In the Ocean of Night, etc.) have FTL? It’s been a while since I read it.
Buddy Holly Is Alive and Well on Ganymede by Bradley Denton.
Howard Waldrop’s collection of short stories Night of the Cooters.
Did they? I’d forgotten. I suppose, novelizations aside, it could just be the same movie magic that allows cross-Atlantic calls without the 10 second delay, though. 
If I remember correctly, the later Rama stuff involved crossing a significant part of the galaxy in 20 years, which would have to be FTL. They’re certainly not worth reading again to check out, though.
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“That being said, enjoy your crappy sci-fi movie.”
Um, that has already been invented. It’s called the undersea cable.
Ura-Maru wrote:
Clarke did stop there.
Rama II, The Garden of Rama, and that fourth Rama book (What was it called? Rama vs. Godzilla or something?) were all written by Gentry Lee.
Sigh. Ok, fine. “it could just be the same movie magic that allowED cross-Atlantic calls without the 10 second delay, BACK IN THE 60’s though.” Happy?
Both Clarke and Lee’s names were on the spine. That makes both of them partially responsible, even if it was obvious that the books were mostly Lee’s. Clarke isn’t that preachy.
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“What, so he becomes a Jedi Knight? Why am I watching this?”
I’d still like to motion for an appeal; you can remove the FTL references from Blade Runner, and the movie wouldn’t change a smidge. It’s just background detail, nothing more.
Actually, a friend of mine suggested that the lack of a noticeable time-lag in the phone conversations with Mars proved that the entire story was, indeed, a dream.
Wow, two whole days and I still have an example no one’s mentioned: Diaspora by Greg Egan. In the course of that novel Earth launches several slower-than-light colony ships ferrying collections of sentient computer algorithms (the protagonists of the story). Not much communication occurs between the colonies, but occasionally a character will literally transmit himself from one to another via maser beam.
Some of Greg Egan’s short fiction takes place in the same universe, for example the story “The Planck Dive”.