No, there is no Planet X. This is just an internet hoax. The facts are:
"Planet X" is a name coined by Lovell for a planet hypothesized to exist but not yet discovered. If it is called "Planet X" then it hasn't yet been found or they would give it another name. The original search for planet X turned up Pluto which is called Pluto, not Planet X.
Up until around 1993, there was a search for a second "Planet X" because Pluto was too light to explain the perturbations in the orbits of Uranus and Neptune.
In 1993, scientists figured out a new mass for Neptune using the Voyager 2 data. With this new information, the discrepancies disappeared. Planet X was no longer needed. The conspiracy sites never mention this.
Harrington, who proposed the new Planet X died in 1993 six months after his planet was disproved, through cancer. A complete coincidence. The conspiracy theorists make a big deal of the way announcements of Planet X searches stopped after his death - this is nothing to do with his death and everything to do with the revised mass of Neptune.
The next search for Planet X was for a companion star called Nemesis which was hypothesized to follow very elliptical orbit and to come closer to the solar system every 26 million years. Never as close as Pluto but close enough to send more comets into the inner solar system. In this theory that event is due to happen next eleven million years from now.
The conspiracy theorists turn that into a star that orbits once every 3600 years, comes right into the inner solar system inside of the orbit of Earth and closer to the sun than Mercury which they say is hidden behind the sun right now - but somehow never spotted on its long journey into the inner solar system. No astronomical paper suggests any of this.
All the way through they accuse NASA of hiding what is really going on. But astronomy is a multi-national activity. Any country of any size has professional and amateur astronomers well capable of observing and plotting the positions of planets and stars. There is no way they could miss a star that came into the inner solar system. It is just absurd. Many exoplanets are found around other stars every year, by many different teams of astronomers, also tiny dwarf planets smaller than Pluto further away than Pluto. Do you think they can find these but can't find a second sun inside the inner solar system?
When Nemesis began to seem increasingly unlikely, next hypothesis was for Tyche in an almost circular orbit well beyond Pluto, the "Good sister" of Nemesis, no increased numbers of comets associated.
In 2014 Nemesis and 'Tyche were both shown not to exist with a very sensitive infrared survey. The conspiracy theorists never touch on this point. Possibly they don't know because a disproof of planet X is less exciting news and doesn't get so widely publicized.
The latest version of Planet X is now a Mars or Earth sized object in a circular orbit about 50 au away - well beyond Pluto to explain a cut off in numbers of icy bodies beyond Pluto called the Kuiper cliff. This version of planet X has not yet been disproved and may exist.
Some scientists also think there may be larger objects further out. There could be a Neptune sized one for instance, but much further away or it would be spotted by now.
In a separate timeline there was a nineteenth century search for a planet called Vulcan in the C19 orbiting permanently between Mercury and the Sun. Though they didn't call it that as this was before Lowell, it is what we would nowadays call a search for a "Planet X".This was shown not to exist and the anomalies in Mercury's orbit explained using General Relativity. There may still be tiny asteroids in between Mercury and the sun of size a few kilometers in diameter called "vulcanoids". A search hasn't turned up any yet though. There is no way our searches could have missed a planet or second star in between Mercury and the Sun.
There are many other absurdities to their claims. I have debunked all the astronomical ones I know of in a blog post on Science20 below. Others have also. Basically these are people who clearly flunked physics at school, and haven't studied basic ideas in astronomy such as orbits and celestial mechanics, and mix up concepts that no astronomer could mix up. They have good imaginations, but probably get most of their ideas of science from the movies.
That basically is the timeline for our astronomical and scientific searches for different versions of "Planet X"
TL:DR - there is no confirmed Planet X - the very name means it is hypothesized but not shown to exist yet. There is a chance of an Earth or Mars sized body in a permanent circular orbit well beyond Pluto, or even a Neptune sized one even further away. This is not disproved. The other versions of Planet X have all been disproved.
Until recently I had no idea anyone takes this Nibiru / Planet X nonsense seriously apart from some of the authors of the books and videos possibly.
But apparently it sounds convincing to many people with not such a strong background in astronomy. So much so that some feel suicidal, or wonder if it is worth doing their Christmas shopping, because they think we will all be gone by Christmas, or are getting treatment for anxiety, unable to sleep, or are very worried and scan the internet every day for the latest "news" on Nibiru via youtube videos and dubious sites like "before its news" where anyone can upload anything with no editorial control.
It is so sad that so many get worried about such absolute balderdash. So I did a short article which had lots of comments by worried people asking many questions and posting links to online articles and videos for me to debunk.
I put much of that material into a long article debunking it all. Then turned it into a kindle booklet. It's actually got a lot of fun astronomy - I spend most of the time talking about the real astronomy behind the Nibiru ideas such as the real searches for planet X. This then continues to get comments with videos and sites for me to debunk, and from time to time I put some of that also into the booklet.