I am asking specifically because I was recently doing a Doctor Who marathon and saw the apalapucia episode and thought "I wonder if that's possible?" So here I am. Is it possible to have a planet where time passes at different speeds at two different locations? Such as: one hour at one place equals a half second in the other.
To the question in the title: No.
To the question in the body text: Yes.
You want a planet orbiting an extremal supermassive Kerr black hole--supermassive so that you can get close to the event horizon without being shredded by tidal forces, and extremal so that the innermost stable orbit is also close to the event horizon. For a non-rotating Schwarzschild black hole, the innermost stable orbit is 3 times farther out than the event horizon, and you don't get particularly strong differential time dilation. But the faster a black hole spins, the closer in the innermost stable orbit gets. If the planet can orbit just outside the event horizon, the side closest to the black hole will experience extreme time dilation, which will fall off fairly rapidly such that time at the opposite "pole" runs noticeably faster.
Such a planet would not actually need a sun, either--the black hole acts as an equally-effective "anti-sun". Rather than having a fairly concentrated heat source (sun) and radiating waste heat into a cold background sky, an extremal black hole planet would be warmed by the blue-shifted background radiation of space, and would radiate waste heat into the cold black hole--which, due to gravitational lensing, would cover an even larger portion of the sky than you would naively assume just from being really close to it.
We already live on such a planet. Welcome to Earth, and more specifically to the Large Hadron Collider (LHC) and other similar places, where particles, accelerated to a significant portion of c, experience the time dilation forecast by the theory of relativity.
The protons each have an energy of 6.5 TeV, giving a total collision energy of 13 TeV. At this energy the protons have a Lorentz factor of about 6,930 and move at about 0.999999990 c, or about 3.1 m/s (11 km/h) slower than the speed of light (c).
Oh, it also happens in the high atmosphere, where elementary particles generated by the nuclear interactions experience relativist effects.
If you want science-based, the answer is a clear NO as far as early 21st century science is concerned. The only things we know to "distort" the flow of time are very high speeds and very high masses -- very high on an astronomical scale. This cannot happen on a planet.
If time has a second dimension (moving from a timeline to a time-plane) then yes, as you can have a variety of time parabolas moving through the time plane at angles to the main timeline, thus giving whatever moves through these other timelines time dilation, which can be extended or contracted.
Some links if you want: https://phys.org/news/2007-05-two-time-universe-physicist-explores-dimension.html