Yes. Genes and DNA are not the same thing. DNA is a means of recording and passing along genes. And alien organism that doesn't use our system of DNA/RNA could still possess genes. So it is theoretically possible to remove all manifestations of genes and have the organism continue to function.

This is not very far fetched. Much of our DNA is used for a short time during development and then effectively turns off for the rest of our life. Taking that DNA away might go unnoticed.