The simplest way to have a region of a planet where it is always daytime, as you point out, is to have the planet be tidally locked to its star. However, you've explicitly excluded this in your question.
I can see two general categories of answers: those with one light source that is active all the time (which probably isn't a star, so the phrase "after the sun has set" doesn't make much sense), and those with two or more light sources (one of which may be a star) that take turns illuminating the region.
My first thought on reading this question was that there may be some kind of onmidirectional, ambient light source completely surrounding your planet. Maybe the planet is buried somewhere in the depths of a very bright nebula, lit up by newborn stars. Or a dying star, for that matter. Maybe the planet is in the middle of a very dense globular cluster, and the light comes from the millions of stars in the cluster- although you might have to explain why none of those stars are close enough to be considered "the sun". Or perhaps the light comes not from a star, but rather a black hole. A quasar's beam could illuminate a dust cloud floating in intergalactic space, which could in turn contain your planet, just as in the nebula example above.
Another possibility is that the light comes not from a star or galaxy, but rather from the universe itself. Very early in the universe's history, everything was very hot, and radiation- light- was simply present everywhere. This primordial light could be what illuminates your planet, perhaps even before the universe cooled to the point where it became transparent and what existed of that omnipresent primordial light became the Cosmic Microwave Background. It wasn't in the microwave spectrum back then, of course (it was in the visible spectrum- the orange part, to be specific), but in the billions of years since, the expansion of the universe stretched out the waves of light, causing it to redshift and cool.
Alternatively, your universe could be heading toward a Big Crunch. When the entire universe contracts, the CMB will heat up again, recreating the conditions in the early universe. Only with more entropy. And more black holes. Probably.
One thing to keep in mind is that in order to function, life (and any interesting thermodynamic process) requires not just energy, but rather an energy gradient. Life on Earth takes in visible light from the Sun, and re-emits infrared back out into space. However, bathing a planet in blackbody radiation from all directions will just heat it up to the same temperature as the radiation, and all thermodynamic processes on it will cease.
This is a problem for the cases where the CMB is the source of light, since that is a blackbody spectrum. However, astrophysics provides a solution: Black holes. Black holes do not heat up when exposed to radiation. In fact, their temperatures depend only on their size- and larger black holes are cooler. A civilization in the early universe could take energy from the CMB and dump its waste heat into a large primordial black hole; whereas one in the Big Crunch era could do the same with either a stellar-mass or supermassive black hole.
The nebula cases don't have this problem, by the way. A planet in an emission nebula is only bathed with a few different wavelengths of light, and could emit waste heat at other wavelengths. For reflection nebulae, it's even simpler: these nebulae mostly scatter blue light, while they're more transparent to red light. And in the globular cluster case, there will be plenty of space between the stars where waste heat could be emitted.
Another, much more mundane possibility, first mentioned in the comments (credit to SRM on this) would be simply be a planet with a very thick atmosphere that scatters light all the way around the planet's surface, causing the entire planet to be in a perpetual state of twilight. It'll be brighter when the star is closer to its zenith, and darker at nadir, but if the atmosphere is thick and dusty enough, there could be enough light to see all day long.
Or, as in Slarty's answer, just put your region at one of the poles of a planet with no axial tilt, and have perpetual dusk.
But those are boring. On to Category 2!
One way to have perpetual daylight on a planet would be to have the planet be illuminated from both sides. Perhaps the planet actually a moon of a large gas giant, and lies at the moon's L1 Lagrange point. After sunset, the gas giant rises, and provides a reasonable amount of light. Or, better yet, the planet could be situated at the L1 point of a binary star system. Perhaps one star is a red dwarf, and the other is a yellow main-sequence star, like our Sun. As the red dwarf is smaller and lighter than the Sun-like star, the L1 point will be closer to it- so the two stars may actually appear to have the same luminosity. In other words, midday when the yellow star is up would be just as bright as midday when the red star is up.
However, the problem with these two scenarios is that the L1 point is a point of unstable equilibrium. In order to actually stay there, your planet will have to actively fire rocket engines to keep from falling into a chaotic orbit. A manmade satellite can do that, for a while, at least, but a naturally-occuring planet will not.
Another way might be to take something like the Earth-Moon system and throw in several more, brighter moons. Earth's moon looks white, but its regolith is actually kind of a dark gray. Changing the material to something more reflective could make moonlit nights considerably brighter. And adding more moons will allow more nights to be moonlit. Now, if the moons are close in to the planet, they will gravitationally interact with each other and potentially destabilize each others' orbits, and there's still the possibility that they might occasionally all be below the horizon at the same time. However, we can solve both of these problems with one stone: Orbital Resonance.
There are several moons in our Solar System whose orbital periods are some simple fraction of that of another moon orbiting the same planet. For instance, three of Jupiter's moons are in a 1:2:4 orbital resonance, meaning for every time that Ganymede completes an orbit around Jupiter, Europa completes exactly two orbits, and Io completes exactly four. This particular configuration doesn't guarantee that there's always a moon above the horizon, but with enough moons, it should be possible to arrange such a configuration. I'm just not sure how.
Alternatively, there could be only one moon in a geostationary orbit. Or, equivalently, the planet could be tidally locked with the moon. This ensures that the moon will always be visible from (and able to provide moonlight to) half of the planet. Your region could be situated directly under the moon.
Another possibility is that the secondary light source is entirely artificial. During the day, the sun shines; during the night, various lampposts and whatnot turn on and provide illumination. Maybe they're solar powered and charge up during the day; maybe the light is actually sunlight, brought over from the other side of the planet by way of fiber optics.