All the time. I had this stray years ago, and he chased me around the apartment until I stood still. He then dropped the dead rat at my feet. This is a tribute, so if that's what they were doing, they love you :)
Just grab some paper towels and use them to toss the little varmints in the outside trashbin. No problemo :)
Our male cat brought us a rodent once, and his sister, Boots, would go into neighbors' garages and bring us small toys or what she could find there. One time she brought us a big leaf and dropped it at our feet and looked up at us. That was her "catch." She seemed so proud.
They're both over the Rainbow Bridge now, but we have good memories.