Does superluminal travel imply travelling back in time? Is this statement true?

You can not travel back in time. If I travel fast enough the clock will start to go backwards, but that does not mean I am traveling back in time. It would only mean that the time reference is producing a negative count. 

 A: One cannot travel backwards in time.
The physics data we have up to now agree completely with the special theory of relativity. This means you cannot travel faster than the velocity of light, and that you can only approach  it as a limit. You will not notice a change in your clock in any measurable way within your travelling system. 
A: Yes, this would be only your impression. E.g. when you are moving away from me then your clock is going slower from my point of view and my is going slower from your point of view. This is because as you are moving away from me the distance between us increases and the light needs longer time to reach you (and this statement is obviously symmetric when we swap me and you). Consequently, the image you'll see of me will be distorted in time and I'll appear to be doing everything slower.
If you would go at the speed of light relative to me (note that this is not physically achievable, but we can discuss this purely theoretically) then all the light from me that could reach you is originating at one moment in my history and consequently it would appear to you that my clock has stopped completely.
Finally, if you were moving superluminally, the light that I emitted as we met and any light I will emit in the future will never reach you (it is simply too slow compared to your speed). On the other hand, as you are moving away, the light that I emitted in the past (and that has already travelled some distance) does have a chance of catching up you. And the further away you move, the older light from me you'll see. Your impression will be that my clock is going backwards. But this is not the end of the story. Because if you were approaching me at the superluminal speeds then it would appear to you that my clock is going forward as usual. Until the point when our wordlines meet. Here's a quick sketch.

The dashed line is you (the superluminal observer), the vertical line is the me (the standing observer) and the 45 degrees lines are light rays I emitted at various events. Note that you'll see me first travelling forward in time upto the event labeled 6 and from the event labeled 7 onwards you'll see me going back in to my past.
A: Spacetime diagram showing how faster than light travel would allow traveling back in time
Since that would very directly break causality, this is a great argument as to why you can't do it.
So I tried to make a diagram to clarify to myself more precisely how that would go like:

SVG source.
Legend:

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*points:

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*A: Spaceship departs from Earth moving towards C slower than light. Note that the spaceship trajectory from A to C coincides with its time axis (because x = 0 at all times) and is not otherwise indicated.

*B: Earth sends a tachyon (fantasy particle that travels faster than light) to C. This moves faster than light (inclination between that of the two black light rays) but it appears to move forward in time according to Earth (stays above its blue space axis)

*C: The tachyon sent from Earth reaches the spaceship, and then the spaceship immediately sends a tachyon back to D. Once again, it appears to move forward in time according to the spaceship (it stays above the red space axis of the spaceship).

*D: The tachyon send back from the spaceship reaches Earth before Earth sent the first tachyon



*lines:

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*black: light rays from B and C

*blue: space and time axes of Earth at B

*red: space and time axes of the spaceship at C

*magenta: tachyons sent from B to C and from C to D



So we see that if we were able to send particles faster than light, even if they appear to be moving forward in time to us, it would be possible to send messages back in time (to a timelike spacetime point) from another inertial frame of reference.
Earth observes the spaceship tachyon travel backwards in time from C to D all the way. This is possible due to the fundamental assumption that the laws of Physics are the same for all inertial frames. So if Earth can send tachyons "forward in time according to Earth", then the spaceship must be also able to send tachyons "forward in time according to the spaceship".
Conversely, the spaceship observes that C happened before B, i.e. it sees the tachyon from Earch move backward in time. This is already a simpler causality problem to observe, but I wanted to show the full "send message to past" in a single diagram. To see that simpler causality issue on the diagram, you just have to add a constant time line (parallel to x of the spaceship), this is done for example at: http://www.physicsmatt.com/blog/2016/8/25/why-ftl-implies-time-travel
One way to explain why Earth and the spaceship don't agree if certain tachyons are moving forward or backwards in time is that the tachyon events are spacelike separated, and therefore can be observed in different orders by different observers.
Variants of the story to make things more dramatic:

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*Earth has a tachyon detector which detonates a nuclear bomb and pulverizes the tachyon sending facility when a tachyon is received. So how can the tachyon be sent, if the facility got nuked in the past?

*instead of sending a tachyon, you send the entire scientist who pressed the send tachyon button around. They then meet themselves.

I later saw this over-the-top bling bling video with a similar diagram: https://youtu.be/an0M-wcHw5A?t=1075 Why Going Faster-Than-Light Leads to Time Paradoxes by Cool Worlds (2022)
