If the buyer has opened a case saying the item has not been received, you will need to refund them unless you have tracking proving delivery.
I sell jewellery - anything over £10 goes tracked. However, in my experience buyers are mainly honest.
I had ONE item not received case last year and none in 2025.
And, just my opinion, sending 5 messages to your buyer is excessive. I would send one saying 'I am really sorry your brooch hasn't arrived. Can you let me know if it does turn up, please?' A disappointed buyer who feels hassled is more likely to leave negative feedback.
"There are two ways to be fooled. One is to believe what isn't true; the other is to refuse to believe what is true.”
Søren Kierkegaard, Danish philosopher (1813 - 1855)