Tidally locked except—not quite. Some precession of the world's core imparts a tiny rotation. It does have a day, but it's two thousand years long.
Each year, 20 leagues of dawnlands are touched by the rays of the sun. Behind them, 20 leagues are claimed by twilight.
Beacuse of this, every living thing slowly migrates. Moving east is an instinct in the soul of every rodent, bird, and people. The giant trees, lichens, and the seething legion of insects beneath the soil stay where they are, but everything else gradually resettles.
It goes on like this for billions of years. The early people were by necessity nomadic, following the dawn. Early tribes would sometimes flourish and expand, taking over swaths of territory and building permanent settlements.
These early kingdoms would appear successful (and through the development of writing would contribute disproportionately to the myths of the world), but each would end in disaster. Torn apart by civil war and chaos as twilight upended their short-sighted arrangements, or sailing stubbornly into the night to be devoured by the things of the dark.
Adventures in the World of Creeping Dark
1. The east is a frenzy of exploration and settlement. Lush new lands are opening up, and the dawnland kingdoms are competing for control of the islands, new trade routes, and a defensible positions for fortresses. There is good money for adventurers, mercenaries.
2. For the truly desperate, ranging far into the pre-dawn nightlands offers the possibility of a precious discovery that could be sold as information, or perhaps even claimed and fortified.
3. Some of these footholds in the pre-dawn night take hold, but to stay viable they must be supplied with news, food, medicines, and specialists.
4. Fighting the vestiges of night. Day comes last to the the valleys, crevices and ravines of the dawnlands. For decades they are inky pools of shadow, and the terrors of the night-side lurk there. Before the lands can be safely settled, these must be made safe.
5. Some of these cannot be disloged, but must be bargained with, placated—or sated.
6. In the far west, the inevitable abandonment of settlements happens unevenly. The day-dwellers rarely give up before they have to, as there are resources that are hard to give up. Mines, fertile lagoons, a final harvest or two of the century crops—all would bring much-needed cash and supplies for the great migration eastwards. But sometimes the dark surges forward - waves of monsters, or lone horrors come soon. These must be fought off, chased back into the night, or at kept at bay while people evacuate.
7. Some things simply cannot be moved, and forays into the darkness to reach them are sometimes necessary. The great fortress of Ing, where the life-giving fountain splashes. It may be a hundred miles into the darklands, but it is the only known cure for the Duke's white palsy.
8. Though they don't like to speak of it, sages have figured out that the daylands are getting narrower. (Court administrators have noticed this too, for that matter.) According to legend, it was once half the world, as you might expect if the world was a sphere. Now, geographers agree: it's a narrow strip, scarcely 800 leagues across. The useful period of human settlements is now barely 40 years! If this waning continues, the people of the day might soon be completely eclipsed.
Sooths from nine kingdoms all claim the same vision—there is a sorcerous cabal in the darklands, the princes of night. Their great working is shrinking the day. They must be stopped, if they can be reached at all.
9. The Duke believes that they have agents on the day-side, moving among us, but this is obviously just the ravings of a man stricken with the white palsy.
EDIT!
This is apparently a recurring idea in fantasy, so there are more resources if you're interested in this.
Scott Charlton was apparently developing the idea at exactly the same day as I was..
Dave McGrogan's The City Standing like a Candle in the Night.
Jahmal Brown has written a full setting based on this idea, apparently to be published through Evil Hat.
Gareth Wilson points out that there's a (non-fantasy) book by Dave Duncan, 'West of January' that is based on a similar idea.
EDIT!
This is apparently a recurring idea in fantasy, so there are more resources if you're interested in this.
Scott Charlton was apparently developing the idea at exactly the same day as I was..
Dave McGrogan's The City Standing like a Candle in the Night.
Jahmal Brown has written a full setting based on this idea, apparently to be published through Evil Hat.
Gareth Wilson points out that there's a (non-fantasy) book by Dave Duncan, 'West of January' that is based on a similar idea.