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Preston of the Spaceways, season 4 episode 11 "Peril in Paradise". Original broadcast 20 June 1968.

Synopsis

The Griffon is dispatched by Adm. Tankersley to Damalia to protect Imperial interests. Along the way (Engineer Patrick) Murphy reports that the spinal chaser mounts are acting up badly (the clash with Dauphine two episodes previously having seriously damaged her, and with not all the damage thus far made good) and (Commander Harold) Preston determines to avoid using them unless it proves absolutely necessary.

Damalia is a Protectorate planet, with a sunny if changeable climate (most of the episode was shot on location in Bermuda, with the naval base standing in for the Resident's palace) and everything at first seems fine, with Preston taking some much-needed shore leave and leaving (Lt. Commander Elizabeth) Clyde in command.
It quickly becomes clear (through scenes viewed by (Gunnery Lt. Gareth) Armstrong on shore leave coupled with further information that none of the core cast is privy to) that there is much resentment over the Imperial interests on Damalia. The middle section of the episode involves a sudden revolt and the capture of Preston (with several casualties on the part of the revolters) followed by an uneasy standoff.
Clyde dispatches the Griffon's gunboat under Murphy's command to attempt a rescue, but the gunboat takes heavy damage from navy-grade surface-to-space missiles and is forced to make a hard landing some miles outside the Protector's palace as the fighting heads towards it.
Armstrong escapes from his captors and makes it to the gunboat (featuring a running gun battle filmed across St George's Island and the airbase) and mans the main pivot gun as they lift off to dock with Griffon - which comes down into atmosphere to pick them up, fighting off surface-to-space missiles and ending up taking further damage. Once docked, Murphy and Armstrong work together to lay the spinal guns manually and destroy the remaining silos (resulting in one of the two active guns burning out).
Griffon orders the surrender of the rebels or a bombardment will take place, and after an argument Preston convinces the rebels to back down.

The denouement to the episode has the Resident making clear that he will work to improve the situation, and Preston files a report on the matter. Among the questions left unresolved by the episode are why Griffon was sent out with battle damage by Adm. Tankersley, and whether the captain and first officer feel they can still work together - as well as where the Damalia rebels got navy antishipping missiles. All of these tie into the next two-parter, "All The Way To The Top".




(Sorry, I seem to have provided something from about a hundred and ten years after the point we've currently got to...)