Holidaymakers and local people on the Isle of Bute have told of their shock and disbelief after police confirmed that six-year-old Alesha MacPhail, who was found dead on Monday morning, had been murdered.

The child, from Airdrie in Lanarkshire had been staying with her father, Rab MacPhail, at her grandparents’ seafront home near the ferry port of Rothesay during the first few days of a three-week summer break.

Her grandmother, Angela King, reported Alesha missing on Monday at 6.25am, posting an appeal for help to search for her on Facebook.

Alesha’s body was found by a member of the public at about 9am in the grounds of a disused hotel about half a mile away.

After warnings from the local police commander to look out for their families and secure their homes, the overwhelming assumption among residents was that the police believed the killer was still on the island.

On Wednesday morning the normally busy seafront was quieter than usual as the community of about 7,000 absorbed the announcement made by police late on Tuesday night that they were now hunting Alesha’s killer.

Locals spoke of their horror at a child being murdered, and in this family resort.

Raymond Yost, whose grandson Aiden played with Alesha on her previous visit to the island, said people were devastated. Showing some Facebook snaps of the grinning pair, Yost told the Guardian: “The whole island is devastated and folk are not letting their kids out of their sight until they find who is responsible. It’s a small community. We’ve all grown up together, went to school together, been out in the world and then came back and it’s heartbreaking to have this happen.”

Praising the police he added that he hoped the obvious presence of the force would help people feel safer. Yost, whose Bonnie Clyde cafe faces the ferry terminal, reported that officers had been scrupulous in interviewing and noting the details of cars and boats leaving the island on Monday.

Throughout Wednesday the house from which the girl disappeared and the wooded area where she was found, were the focus of intense police activity.

Police confirmed on Wednesday afternoon that a vehicle had been recovered as part of the investigation. Local officers are being assisted in their investigations by personnel drafted in from across Argyll and Bute and West Dunbartonshire divisions, as well as by forensic specialists.

Officers at Rothesay police station, which remains open 24 hours a day unlike stations on many small islands, said no crime like this had happened on the island within their working memory, describing Bute as very safe with remarkably low crime rates.

Confirming that Alesha had been murdered, DS Stuart Houston, from Police Scotland’s major investigation team, refused to reveal the cause of death or to confirm where police believe she was killed.

Speaking on Bute on Tuesday night Houston described Alesha’s family as “utterly devastated” and said he was keen to speak to anyone who was involved in searching for the girl in the early hours of Monday.

Referring to rumours that Alesha could have sleepwalked out of her home, he said detectives were exploring a number of lines of inquiry.

Flowers, cards and cuddly toys were left outside the house on Ardbeg Road, one of a series of detached villas facing directly across the water to Wemys Bay, where Alesha’s grandmother lived in the top-floor flat.

Alesha’s father, Rab MacPhail, 25, and mother, Georgina Lochrane, 23, who are separated, were both distressed as they read through the messages left by well-wishers.

Lochrane appeared to find out about her daughter’s disappearance from reading the Facebook appeal, leaving a series of increasingly desperate posts, such as “Angela answer me now”.

Rab MacPhail’s brother Calum later responded to criticism, also posting on Facebook that his parents had tried to contact Lochrane before she found out the news through other means.