The late singer's love has been spotted for the first time since the One Direction alum fell to his death in Buenos Aires, Argentina last week ... and the influencer appears to be keeping a low profile in the wake of the tragedy.

Kate, who had been dating Liam for about 2 years before his death, was seen out and about in Florida Sunday, where she ran an errand to grab some dog food not far from her Palm Beach abode.

Kate appeared somber during the outing ... choosing to wear only gray sweatpants, slides, and an oversized tee while accompanied by her and Liam's beloved dog.

The TikTok star addressed Liam's passing in a poignant social media note in the days after his death. In a message to her followers, she thanked fans for their support and love ... noting she was at a "complete loss" by the news.

As she continued, she praised Liam as her "angel" and shared she loved the singer "unconditionally."

Kate had notably been staying with Liam in Argentina in the days leading up to his fatal fall from his hotel balcony. As TMZ previously reported, Kate had decided to return to Florida ahead of Liam ... revealing she was ready to be home after their 5-day trip extended to a 2-week long getaway.

Police have said that Liam jumped to his death, indicating a possibly hallucinogenic drug may've been a factor in the tragedy ... but are waiting for test results on the substances found in his room.