Does the body keep trying to heal itself after death?

You need blood flow to continue healing. It carries oxygen, in the blood, needed for healing.

No

No as the others have said.

Negative.

It would try, but it would fail.

Death is final unless you awaken to find bitemarks on your rotting fingers and body in which case the zombie apocalypse has begun.

No. Everything ceases. What would it’s purpose be if it did?

As others here have said, without blood flow tissues cannot heal. However, it is my understanding that hair, and finger and toenails do continue to grow for a short time after the body dies.

^^ This is a common misconception. What actually happens is that the skin around the hair and nails dehydrates and retracts, giving the appearance of growth. The growth of hair and nails requires active cell division, which stops when you die.

Depends on your definition of death.
I think that if there is no cerebral brain activity then Elvis has left the building and consider that death (yet the medulla oblongata may be fully functional and continue to keep heart and lungs functioning). In such a case the body will continue to try to heal itself.

If the heart stops functioning then all other organs will quickly stop functioning.