It's definitely mentioned in Prince Caspian — or at least, the place where the Pevensies first arrived in Narnia is, by that time, named Lantern Waste. (That would be "waste" in its older meaning of uninhabited and uncultivated land.) Pattertwig is sent there to see if the four Kings and Queens will reappear there after Caspian blows Susan's horn (while Trumpkin is sent to Cair Paravel, which of course is where the Pevensies do re-enter Narnia).

I don't think we're told specifically whether the lamppost is still standing at that place in Caspian's time, or whether it had disappeared by then and the area of Lantern Waste was still named after it even though the actual lamppost was no longer there. Lantern Waste is also referred to again in LB — it's the area where the guard tower is that Tirian and Jill and Eustace hide out in — but again, no mention of the lamppost itself. (Going from memory here, as I don't have the books with me at present, so am happy to be corrected if I'm wrong!)

We do learn later from MN (which Lewis wrote several years after LWW and PC) that the lamppost is actually a living thing that grew from the iron bar that Jadis threw (which she'd wrenched off a lamppost — probably a gas lamp — in late Victorian London). Going by the timeline that Lewis apparently compiled after he'd written all seven books, the lamppost is already 1,000 years old when Lucy first arrives in Narnia, and then another 1,303 Narnian years pass between then and the four Pevensies coming back in Caspian's time. Maybe, like a tree, the living lamppost simply reached the end of its natural life and died some time during that interval?? That's the most logical explanation I can think of for why the lamppost itself is never mentioned again. But it doesn't seem to be something that Lewis thought out specifically.

Oh, don't we just love to speculate about related Narnia things?

Given that The Magician's Nephew is set during late Victorian era, the lamppost almost makes me think of what is used for a setting of Charles Dickens A Christmas Carol (and you know how many times that's been done).

It's true that we're not told whether the Lamppost was still standing or if it died eventually. I would think it would still have to be there in The Horse And His Boy, because it takes place during The Lion, The Witch And The Wardrobe (according to the Timeline, about a year before the Pevensies hunt for the White Stag and vanish back to our world Narnian time).

It could have disappeared when the Telmarines took over. They have wanted to wipe out the memories of Old Narnia and pretend that it never happened.

Posted by: LWW

"By the Lion's Mane, a strange device," said King Peter, "to set a lantern here where the trees cluster so thick about it and so high above it that if it were lit it should give light to no man!"
(my emphasis)
[...]
So these Kings and Queens entered the thicket, and before they had gone a score of paces they all remembered that the thing they had seen was called a lamppost,

When Lucy first entered Narnia, she could see the light from the lamppost and then walk for ten minutes to reach it.

Now it is not lit any more.

This shows that by the end of their reign in the Golden Age, the lamppost had lost its light, but was still standing. Perhaps this meant that as a living iron creature it had now died, and would start rusting away. The name Lantern Waste could keep, though, whether the memory of the actual lantern had been kept or not.

I wonder if anyone would want a story about what may have happened to the Lamppost? Perhaps, for a student, write something about it for a school project.

Some may even want to write an essay about the Lamppost itself. I remember when I was in my freshman year of High School, I wrote an essay about the Dawn Treader, which described the ship and what kind of role it played. 

So for you students that are reading this, perhaps you can write an essay about the Lamppost, describing it, and what kind of role it played or even a story about what you think may have happened to it after the events of LWW.

I would speculate that the light of the lamppost had to be lit, and that it was lit and maintained during the witch's reign for some reason (like a beacon for smugglers, perhaps) and when her reign was over, there was no longer a reason for it to be lit, and no longer a reason for creatures to pass through that part of Narnia any more, hence the overgrowth of trees.

I feel like I wrote this in another topic, but I can't find it now, so here goes.

I don't really get why readers would expect the lamppost to appear in more books. I mean, even in the ones it does appear, it doesn't do much. I love how C. S. Lewis explores new territory in different books. (We get the sea in The Voyage of the Dawn Treader, Ettinsmoor in The Silver Chair, Archenland and Calormen in The Horse and his Boy.) I see no reason why he should have had the characters hang around Lantern Waste, which we'd already read about. 

If the lamppost was still alive by the time of LB, I imagine it flickering out when Narnia comes to an end.