The thing is, Everstones are hold items; at least in the games, they only work as long as the Pokémon is holding them. When the Everstone is removed, the Pokémon that was holding it will evolve at the first opportunity. They seem to function like puberty blockers in that sense (at least, if my understanding of puberty blockers is correct).
The anime isn't necessarily beholden to the games' logic, but there is typically some level of consistency, especially by the DP series. While Brock states that "an Everstone has the power to prevent Piplup from evolving, ever," this does not necessarily mean it does not have the condition that Piplup needs to keep holding it. Assuming this is the case, Piplup would need to keep the Everstone on him pretty much constantly, because his body would be constantly trying to evolve without its influence; over the course of DP122, which takes place across roughly 24 hours in-universe, he needs to use Bide to suppress his evolution 5 times: once the night before the bulk of the episode takes place, once during the daytime, then three times that night. The last of these suppressed evolutions occurs at an incredibly inconvenient time (during a scuffle with Team Rocket) and lasts for well over a minute of the episode's playtime, in contrast to the prior ones that only lasted a few seconds. Even if some of the events that are shown happening within those 80 seconds or so are happening concurrently in-universe, it still can't be much shorter than a minute in-universe. (There is also an instance of Piplup using Bide for its intended purpose of absorbing and paying back damage, but this probably was not also used to suppress evolution.) And these are just the instances we, the viewers, see; in both the first and fourth instances we see in the episode, Piplup is sitting under a waterfall, as though he was anticipating the urge to evolve, and after the third instance, Brock says, based on what he's inferred so far, that "Piplup's been at this [suppressing his evolution] for a while." The five instances we see are not the only times Piplup's had to suppress his evolution, what we see in this episode is just where everything comes to a head.
But after he gets the Everstone, we never see Piplup use Bide to suppress his evolution again, so he obviously is still under the Everstone's influence. And the Everstone's influence would have to be constant to counteract his body's need to evolve, given the events of DP122. So where does Piplup keep it?
Now, the logistical issue of "how does a Pokémon hold an item for an extended period of time" is present in the games as well, but that's pretty easy to handwave for the sake of gameplay. However, this kind of handwaving is a bit trickier to do for Piplup's case. It's not like the Everstone is just chilling in Dawn's bag or something like that; if it did work to suppress Piplup's evolution like that, then that raises a whole lot of questions like "what happens if Piplup gets too far away from Dawn?" or "how does that Everstone not also suppress the evolutions of a bunch of other Pokémon, like Ash's Torterra, while still suppressing Piplup's evolution?" And again, by the time Piplup is given the Everstone, he is needing to suppress the his evolution multiple times a day and at inconvenient moments, so it's highly unlikely that Piplup is only being exposed to the Everstone periodically, unless the Everstone's effect is residual in some way, which would contradict how the Everstone works in the games. Which, again, not necessarily a deal breaker, but it definitely makes this theory a hard sell. This means that Piplup would have to have it concealed on his person.
This might sound implausible at first; Piplup doesn't exactly have pockets. But I would argue that perhaps he has something similar. The Pokédex states that Piplup has "thick down that guards it from the cold," much like real penguins. I could easily see it being kept in somewhere in his down, perhaps in a locket or satchel he keeps around his neck. The Everstone is reasonably sized for that; as we see it in the anime, it could probably fit inside Piplup's beak. (I don't think he would keep it in his beak, though, that would get in the way of his vocalizations and likely his use of moves like Bubble Beam, and just sounds uncomfortable.) In fact, we see another Pokémon do something similar in the anime: the Aura Sphere Riolu from DP071 and DP072 (the two part episode titled "Pokémon Ranger and the Kidnapped Riolu!" in the English dub) carries a memento in a satchel they keep around their neck. And Piplup has the benefit of being able to nestle such a satchel among his down. That might have a few implications for his hydrodynamics, but that's a fairly minor thing in comparison to any other place I can think of that Piplup could keep his Everstone.
ncG1vNJzZmiapaGvorPAq5uepl6jsrV706GpnpmUqHy4tMSrnGacn5rAbrDAsKWsZaCeva3Bz2ainp2gYrWqv4yerZ6qo6m8r7GNbGdpa2JmfA%3D%3D