Is it possible to have post-partum three years after childbirth? Or is it just plain depression?

Depression. I think at this point it is depression. You need to talk to your doctor and family. You may need medication and therapy.
Yes, this is depress. Althought the stress of children continues, at three years of age you should be able to adapt to being a mother and the demands of raising a child. Symptoms this late would be treated as depresion.