The late singer's loved ones issued a statement Thursday, a day after Liam fell from the balcony of his hotel room in Buenos Aires, Argentina and died.

In a note shared with the BBC, Liam's family said they are "heartbroken" following the news of his passing, and are taking time to remember the pop star's personality and legacy.

They wrote ... "Liam will forever live in our hearts and we'll remember him for his kind, funny and brave soul. We are supporting each other the best we can as a family and ask for privacy and space at this awful time."

The Payne family's words come amid fans' devastation over Liam's death in South America ... with countless Directioners standing vigil outside the hotel where he passed.

As you can see in photos, mourners fought back tears and comforted one another as police remained on the scene ... investigating the specifics of Liam's fatal fall.

As TMZ previously reported ... Liam was pronounced dead at the CasaSur Palermo Hotel located in the Palermo neighborhood of Buenos Aires. According to the 911 call, Liam allegedly had a meltdown while intoxicated in his hotel room on the 3rd floor ... where he later fell from the balcony.

He had been in Argentina supporting former One Direction bandmate Niall Horan at his show at the Movistar Arena. As Liam's girlfriend, Kate Cassidy, revealed online ... Payne extended his trip to spend more time in Argentina.