Perhaps they do. Look at the second pilot episode "Where no Man Has Gone Before".
Remember Kirk's log at Stardate 1313.1:
Star date 1313.1. We're now approaching Delta Vega. Course set for a standard orbit. This planet, completely uninhabited, is slightly smaller than Earth. Desolate, but rich in crystal and minerals. Kelso's task, transport down with a repair party, try to regenerate the main engines, save the ship. Our task, transport down a man I've known for fifteen years, and if we're successful, maroon him there.
After transporting Gary down and some stuff that takes a few minutes of screen time and may take a few hours in fictional time:
Captain's log, Star date 1313.3. Note commendations on Lieutenant Kelso and the engineering staff. In orbit above us, the engines of the Enterprise are almost fully regenerated. Balance of the landing party is being transported back up. Mitchell, whatever he's become, keeps changing, growing stronger by the minute.
[Delta Vega Brig]
DEHNER: He's been like that for hours now.
KIRK: Have Doctor Piper meet us in the control room with Kelso. We'll all transport up together.
SPOCK: If he should try to stop us
KIRK: Kelso will be on the destruct button until the last minute. I think he knows that.
DEHNER: I'm staying behind with him.
[Delta Vega Control room]
KELSO: Fission chamber three checks out. The station seems to be running fine.
SCOTT [OC]: You're a talented thief, Kelso. Everything you sent up seems to be fitting in place.
KELSO: I'm kind of proud of the job we've done. We're going to be ready to transport up (a cable loops over his neck from behind and throttles him)
[Delta Vega Brig]
KIRK: You're leaving with the ship, Doctor.
DEHNER: He is not evil.
KIRK: I gave you an order, Doctor.
MITCHELL: You should have killed me while you could, James. Command and compassion is a fool's mixture. (zaps Kirk and Spock then waves away the forcefield. He leads Dehner into his cell and we see in the mirror her eyes are like his now)
Everything after the Captain's log on stardate 1313.3 should take just 2 or 3 minutes.
In the next scene Kirk wakes up.
PIPER: It hit me, too, whatever it was. Kelso is dead, strangled. At least Spock's alive.
KIRK: Doctor Dehner?
PIPER: She went with Mitchell.
KIRK: Don't give him a pill until after I'm gone. It's my fault Mitchell got as far as he did. Did you see their direction?
PIPER: Yes, there was some morning light. They were headed across the valley, to the left of the pointed peaks. There's flatlands beyond.
KIRK: When Mister Spock recovers, you'll both transport up immediately to the Enterprise.
PIPER: But Captain
KIRK: If you have not received a signal from me within twelve hours, you'll proceed at maximum warp to the nearest Earth base with my recommendation that this entire planet be subjected to a lethal concentration of neutron radiation. No protest on this, Mark. That's an order.
So Kirk has to report back within 12 ours or be stranded on Delta Vega.
Later Mitchell makes a tombstone for Kirk and it has an inscription:
James R Kirk
C 1271.1. to 1313.7
MITCHELL: I'm disappointed in you, Elizabeth. (phaser rifle has no effect on him) I've been contemplating the death of an old friend. He deserves a decent burial, at least. (a grave appears in the rock, with a tombstone engraved James R Kirk. A massive boulder on the cliff above is loosened)
DEHNER: Stop it, Gary.
MITCHELL: Morals are for men, not gods.
KIRK: A god, but still driven by human frailty. Do you like what you see?
MITCHELL: Time to pray, Captain. Pray to me.
KIRK: To you? Not to both of you?
MITCHELL: Pray that you die easily.
KIRK: There'll only be one of you in the end. One jealous god. if all this makes a god, or is it making you something else?
MITCHELL: Your last chance, Kirk.
KIRK: Do you like what you see? Absolute power corrupting absolutely. (Dehner zaps Mitchell, he returns fire, she continues and his eyes return to normal)
DEHNER: Hurry. You haven't much time.
(The two men punch and wrestle until Kirk raises a rock over his head)
KIRK: Gary, forgive me.
(Mitchell's eyes glow again)
MITCHELL: For a moment, James, but your moment is fading.
(They both fall into the grave, but Kirk leaps out first, grabs the rifle and brings the boulder down to entomb Mitchell)
DEHNER: I'm sorry. You can't know what it's like to be almost a god.
KIRK: Enterprise from Captain Kirk, come in.
Mitchell expects to kill Kirk on stardate 1313.7, but Kirk kills him and calls the bridge a few minutes later, obviously before 12 hours have passed.
And the next scene is on the bridge.
KIRK: Captain's log, Star date 1313.8. Add to official losses, Doctor Elizabeth Dehner. Be it noted she gave her life in performance of her duty. Lieutenant Commander Gary Mitchell, same notation. I want his service record to end that way. He didn't ask for what happened to him.
SPOCK: I felt for him, too.
KIRK: I believe there's some hope for you after all, Mister Spock.
So from stardate 1313.3 to 1313.7 or 1313.8 is less than 12 hours. So about 0.4 or 0.5 of a stardate unit is less than 12 hours. This indicates that a stardate unit can not be more than 30 hours long.
Captain Kirk makes this log earlier:
Captain's log, Star date 1312.9. Ship's condition, heading back on impulse power only. Main engines burned out. The ship's space warp ability gone. Earth bases which were only days away are now years in the distance. Our overriding question now is what destroyed the Valiant? They lived through the barrier, just as we have. What happened to them after that?
After several more scenes Spock makes a recommendation to Kirk in the briefing room:
SPOCK: We'll never reach an Earth base with him aboard, Jim. You heard the mathematics of it. In a month he'll have as much in common with us as we'd have with a ship full of white mice.
KIRK: I need a recommendation, Spock, not vague warnings.
SPOCK: Recommendation one. There's a planet a few light days away from here. Delta Vega. It has a lithium cracking station. We may be able to adapt some of its power packs to our engines.
KIRK: And if we can't? We'll be trapped in orbit there. We haven't enough power to blast back out.
SPOCK: It is the only possible way to get Mitchell off this ship.
KIRK: If you mean strand Mitchell there, I won't do it. That station is fully automated. There's not a soul on the whole planet. Even the ore ships call only once every twenty years.
SPOCK: Then you have one other choice. Kill Mitchell while you still can.
Kirk decides to go to Delta Vega and the next scene opens with his log on stardate 1313.1 when they are approaching Delta Vega.
If Delta Vega was a few light days from their position (more than two and less then seven which would make a light week, I guess) it should take a light ray two to seven days to reach Delta Vega. If the Enterprise is traveling about 0.1 to 0.9 times the speed of light under impulse power it should take it about 2.222 to 70 days to reach Delta Vega.
But between stardate 1312.9 and 1313.1 a bunch of stuff happens, Kirk decides to go to Delta Vega, and they reach Delta Vega. So the trip to Delta Vega seems to take less than 0.2 stardate units, and thus less than six hours - possibly a lot less than six hours.
Even with the main engines burnt out and operating under impulse power, the Enterprise traveled faster than light to reach Delta Vega much faster than the two to seven days light would take.
Earlier, when discussing the fate of the S.S. Valiant:
SPOCK: Decoding memory banks. I'll try to interpolate. The Valiant had encountered a magnetic space storm and was being swept in this direction.
KIRK: The old impulse engines weren't strong enough.
SPOCK: Swept past this point, about a half light year out of the galaxy, they were thrown clear, turned, and headed back into the galaxy here. I'm not getting it all. The tapes are pretty badly burned. Sounds like the ship had encountered some unknown force. Now, orders, counter orders, repeated urgent requests for information from the ship's computer records for anything concerning ESP in human beings.
That makes it seem that the Valiant used impulse engines to travel much faster than light. Another possibility is that ships of era of the Valiant used the impulse drive at the same time as the warp drive to achieve FTL speeds.
If the trip to Delta Vega took exactly six hours of ship time - which seems to be the maximum possible - and lasted for 2.22 to 70 days outside the ship, time must have been slowed by factor of about 8.8888 to 280 times aboard the Enterprise during the trip to Delta Vega - possibly many times as much if it took less than six hours outside time to reach Delta Vega.
If that difference is caused by relativistic time dilation caused by the speed of the Enterprise under impulse power the Enterprise can certainly reach almost the speed of light under impulse power and have significant relativistic time dilation.
Maybe the Enterprise did not reach Delta Vega Under impulse drive using the impulse engines. Kirk's log said the main engines were out and they lost spqce warp capability and where heading on impulse power only. So maybe they were using hypothetical auxiliary warp drive engines powered by the power source of the impulse engines instead of the power source for the warp drive. So the Enterprise could still go tens or hundreds of times as fast as light, but much slower than the main engines with their space warp capability.
And in "Balance of Terror" they discuss the capabilities of the Romulan ship:
KIRK: Yes, well gentlemen, the question still remains. Can we engage them with a reasonable possibility of victory?
SCOTT: No question. Their power is simple impulse.
KIRK: Meaning we can outrun them?
This may mean either that the Enterprise is faster than the Romulan ship because the Enterprise has warp drive engines and the Romulan ship has only impulse drive engines, or it could mean that the Enterprise and the Romulan ship both have warp engines but the warp engines of the Enterprise have a superior power source and the the warp engines of the Romulan ship still use an inferior power source that the Enterprise only uses for its impulse engines.
The warp drive of the Enterprise uses both a space warp and a time warp to travel faster than light.
Note that in the scenes from the first pilot film "the Cage" Lieutenant Tyler tells the survivors who crashed about 31 years before "The menagerie" that:
TYLER: And you won't believe how fast you can get back. Well the time barrier's been broken. Our new ships can
Tyler obviously means that the travel time either in the outside universe, or aboard the sip, or both is much shorter now that the "time barrier" has been broken.
Earlier Captain Pike addressed the ship:
PIKE [on screen]: Our time warp, factor seven.
And in a scene from "The cage" that was not in "the Menagerie":
SPOCK: This is the acting captain speaking. We have no choice now but to consider the safety of this vessel and the remainder of the crew. We're leaving. All decks prepare for hyperdrive. Time warp factor.
Clearly in the era of "The Cage" the latest generation of warp drive is called hyperdrive and includes the newer time warp capability as well as the older space warp capability. And probably the warp drive in the era of TOS - only 13 years after "The Cage" also includes both the newer time warp capability as well as the older space warp capability.
After encountering the galactic barrier the space warp capability was knocked out but the Enterprise could still use the time warp capability to travel faster than light. Kirk's log on stardate 1312.9 said that "Earth bases which were only days away are now years in the distance". If days means one to seven days (less than weeks), and years means one to ten years (less than decades) the normal speed of the Enterprise would be about 52.178 to 3652.5 times faster than its reduced speed.
If Kirk meant the normal speed was warp factor six and warp factor six is 216 time the speed of light according to the TOS warp formula, the reduced speed of the Enterprise would be about 0.059 to 4.1399 times the speed of light.
If Kirk meant the normal speed was warp factor five and warp factor five is 125 time the speed of light according to the TOS warp formula, the reduced speed of the Enterprise would be about 0.0342 to 2.3956 times the speed of light.
If Kirk meant the normal speed was warp factor four and warp factor four is 64 time the speed of light according to the TOS warp formula, the reduced speed of the Enterprise would be about 0.0175 to 1.2265 times the speed of light.
Thus Kirk's statement seems consistent with the reduced speed of the Enterprise being either 1) sublight and too "slow" to have enough time dilation, 2) sublight but fast enough for enough time dilation to slow down time enough, and 3) just a few times faster than light and needing some type of artificial "time warp" to slow down time aboard the ship enough.
Thus the possibilities seem to be:
1) That ships used impulse drive as early as 200 years before TOS to travel much faster than light, and the Enterprise could still use impulse drive to travel faster than light as late as "Where No Man Has Gone Before". Perhaps early warp drive was called impulse drive.
2) The Enterprise used impulse engines to travel at relativistic speeds to Delta Vega and time dilation slowed down time aboard the ship tens, hundredrs, or thousands of times.
3) The Enterprise used hypothetical auxiliary warp drive engines to travel faster than light to Delta Vega powered by the power source for the impulse drive instead of the offline usual power source for warp drive.
4) The warp drive of the Enterprise uses both a space warp and a time warp. The space warp function was knocked out but the time warp function still worked. Thus the Enterprise may have traveled to Delta Vega much slower than normal and used the time warp function to slow down time aboard the ship enough to get to Delta Vega in just a few hours time aboard the ship, or else used the time warp function to get to Delta Vega faster than light in just a few hours time both inside and outside the ship.
For over forty years Star Trek episodes have been rerun, and scripts, videotapes, and DVDs have been sold, and yet how many technical manuals, etc. have ever suggested any of those four possibilities?