We're told the prosecutor in the case has officially ruled Liam did not die by suicide, stating the effects of the drugs the One Direction alum took caused him to fall to his death from the balcony of his suite at the CasaSur Palermo Hotel back on October 16.

As we told you Thursday, 3 people have been arrested in the case -- a housekeeper at the hotel, a friend of Liam's, and an alleged drug dealer ... this after multiple raids conducted by local police.

TMZ broke the story ... the British singer had been staying at the hotel after his girlfriend, Kate Cassidy, returned to Florida following their extended visit in support of Niall Horan, Liam's former One Direction band mate, who performed a concert at the Movistar Arena 2 weeks earlier.

Initial autopsy results revealed Liam had a variety of drugs in his system at the time of his death ... including "pink cocaine," and "cristal," among others.

Photos from Liam's hotel room confirmed drug paraphernalia was found among his belongings, as well as a suspicious Dove soap box ... which prompted investigators to look into how the drugs came into the pop star's possession.

Liam had previously been open about his addiction struggles, sharing in a YouTube video last year that he had done a 100-day stint in rehab.

As TMZ previously reported ... Liam had been telling people in his immediate circle that he was still sober in the months prior to his passing ... with those close to him noting he seemed in "good spirits."

It's unclear what sparked the lapse in his sobriety ... though, it seems police are placing the blame on those who gave LP the drugs.

Liam, who was 31 at the time of his death, is survived by his parents, Geoff and Karen, sisters Nicola and Ruth, and 7-year-old son, Bear ... who he welcomed with ex Cheryl Cole.