Use the binoculars if you spot one. You'll get the info you need like the best weapon to use to kill it. Also if possible aim for the head to get a perfect pelt.

I’ve tried that, but I always end like cat food

Traps and Bows

My best advice is to ride through the little forest near "Old Harry Fen" in the late morning hours and you will eventually get chased by a Panther. Do NOT dismount your horse, calm it down while riding out of the forest towards the shore (the Panther will continue chasing you), where you have to dismount your horse and quickly climb on a rock. The Panther can't get up there and will slowly walk in front of the rock and just stare at you, giving you the chance to study it and go for a clean headshot. Of course, if you want a perfect pelt, you have to be lucky that the Panther has three stars...

This map might help:

Many thanks! I’ll try it

Just wait until it attacks, then (after the first attack) use a rifle to shoot it in the head.