Is there any part of the human body that does not contain DNA?

If your body produced it, it has DNA.

Think about it, DNA is what instructs your body on how to make something, so naturally it has DNA in it.

The only cells which lack DNA are the mature red blood cells (erythrocytes). This is because they lack a nucleus, which is where the DNA is found in other cells. Red blood cells also lack mitochondria which themselves have DNA. Therefore they not only lack nuclear DNA, but mitochondrial DNA as well.

Red blood cells develop in your bone marrow from special cells called stem cells. These do have a nucleus, but as the red blood cell develops the nucleus is squeezed out. This makes more room for the red pigment hemoglobin, which is needed to carry the oxygen in the red cells.

The outer layer of the skin is covered by cells stuffed with the protein keratin. These dead cells do not have any DNA. Finger nails and hair are also made up of cells stuffed with keratin, and they don't have DNA in them. Sometimes DNA may be present on hair if there are cells attached to the root of the hair. The rest of the hair shaft does not contain DNA. Red blood cells also lack nucleus and DNA.