Instant communication while travelling close to the speed of light... and what about time?

Hi everyone, I've just discovered the series and loved it, I'll probably move to the books now.

First of all, sorry for my English, not a native speaker but I'll do my best.

So, let's accept all the sophons' mechanism to get to instant communication, but having the spaceship moving close to the speed of light wouldn't have as a result time passing much slower on the ship respect to Earth?

If this is the case, the message would move instantly from the Earth to the ship, but the time needed (on the ship) to listen to the message, thinking to an answer, reading the answer out loud, pressing the "send" button and so on... wouldn't all of this result in a lot more waiting time on Earth, actually making instant communication impossible?

Thanks to anyone who will help, as I can't get this dilemma out of my mind! (⌒_⌒;)