Plot hole: The way Jack narrates the murder does not make sense; according to him, the killer went first to grab the telephone, then back to the garden to kill the victim, then from there, he had to go, unnoticed and with his robe loose, back to the shack (which seems to be close to the entrance and the box where the phone was). Moreover, to stick the rope into the clay sculpture, especially the way we see it, not entwined into a ball but unfurled, he needed to disrupt the sculpture in a way that would have been noticeable, even if the clay was not entirely dry yet.
Revealing mistake: Jack searches for Bryn Williams. We see the page with the results. Florence suggests him then to change to William Bryn, he types it, but the next close-up shows the results page with the query unchanged from before. Still, the bottom result (the top one are still all "Bryn Williams) is the one ("William Bryn") they are looking for. (00:16:20)
Continuity mistake: At the harbour, Florence's ponytail is suddenly windblown in the shot when Jack says "I'd say that gives you a pretty good reason to want revenge." (00:37:15)
Continuity mistake: Officer J.P. Hooper holds the wad of bills for an unnaturally long amount of time, but if it was for sake of continuity, too bad; he holds it with his arm at a different angle between shots (he did not have his elbow on the desk, and all of a sudden he does). (00:37:20)
Continuity mistake: The detectives reach Cressida Friend on the beach. Jack waves around the letters with his right hand; his left hand is in his pocket in the front views, but it's on the waist in the wider shot. (00:32:00)
Continuity mistake: JP presents the key after his visit to Dread Locks; Tobi Bakare is holding the plastic bag in a different way throughout the scene (just one finger in the reverse shots) and at towards the end he lowers his hand in a shot but it's again up in the next. (00:27:10)
Plot hole: One of the guests is a journalist going undercover. The police finds him out because...they google his name, "Bryn Williams journalist" (no quotation marks), but then not finding anything (and Jack says so despite actual results being visualized, but he dismisses them at a glance) Florence has a stroke of genius and says "William Bryn then." And this time the googling pays off, with a search result page that says, literally, that he's an investigative journalist "known for going undercover to investigate", apparently being a master at that having won prizes! In all this amazing silliness, it appears quite impossible that they wouldn't know his real name, since they already identified the suspects and ran background checks, which in every episode always include checking with immigration when they entered Saint Marie. He couldn't have entered the country under a fake name.
Continuity mistake: When Jack speaks to Florence about his ambiguous intention to spend some quality time with the Commissioner, he changes position in between shots, with his excited fist raised while his left hand disappears from sight. (00:15:10)
Continuity mistake: After the intro, Florence reaches Jack at the shack and leans against the wall while he's whining about the speech he has to write about their boss. In all close-ups Joséphine Jobert's hair flows in front of her shoulder, but in the one wider angle (when she shrugs him off with "I don't know, sir") her hair is fully behind her back. (00:04:25)
Other mistake: According to his passport, Michael Bennett was born on July 120th. (00:35:50)
Answer: Most likely they had only tropical exotic drinks, he wanted an old fashioned English beer.