I received an item after I requested a refund, can I keep it?

It seems odd to even think about this until after it happened...

But if it does happen, then the right thing to do would be to set it aside, and contact them again. If after a while (perhaps a couple of weeks, or maybe a month) you don't hear back from them and they don't pick it up, then I suppose it would be OK to assume you can keep it. How long you should wait is certainly subjective, and may depend on the value of the item. If it's valued at $500 I'd hold it longer than if it is worth $50.

Note if you do keep it, and they eventually contact you about the item in the future, I would be prepared to either purchase or return at the later time, if you still have it.

Under US law, you're entitled to keep unsolicited merchandise and not pay for it. Whether you should or not is up to you and your conscience.

About twenty years ago I bought an expensive remote control from an online retailer. Worked fine. Six months later they sent me another one. Followed by an e-mail -- they had updated their order-management software, and somehow ended up sending out hundreds of duplicates of previously-filled orders. They pointed out that I was entitled to keep it, and that they were not entitled to demand payment. If I didn't respond, that would be the end of the matter. But if I was willing to send it back to them, they'd arrange return shipping and give me a credit for 10% of its value. I sent it back, and didn't use the credit.