Does yesterday have buddha nature I am asking because I think that if the past has buddha nature, and the dharmakaya is impermanent, then there is nothing left of the enlightened when they die.
Dharmakaya is not impermanent. Where did you get such an idea?
Buddha nature/dharmakaya is not impermanent. Dharmakaya = the idea of Self, Buddhanature, Allah, Brahmin, Substrate... it's the thing that is beyond existence and nonexistence... it's beyond separation and non-separation. It's the thing that stands beyond time and exists without existing.

"Past" is a way of categorizing time.
It is just as much a figment of imagination as "present" and "future".
The only thing that exists is the "now" but even that is speaking relative to a "past" and "future" which do not truly exist except in your imagination.
Nonetheless, the imagination is an important tool for discussion, desire, economics, etc. It is the foundation of our worldy and supernatural discoveries.

Anyway, to give you a clear answer to your unclear question, the past does not have buddha nature since it is completely imaginary, dharmakaya is permanent (it is all-that-is), and your logic does not follow about enlightened people.

On the other hand, it is true, much is left behind through the bodhisattva path including the fetters which bespeaks of the fear you have of "nothing being left".