Can eyelid swelling during pregnancy cause permanently stretched skin?

The swelling of pregnancy can cause the body and face to swell significantly.  The swelling is very unlikely to be severe enough to stretch the skin permanently.  It can take a few months after the delivery for all the swelling to resolve.  

In my experience after pregnancy and resolution of the condition these findings usually improve.  

don't worry! You should fully recover but it may take a couple of months.  Fluid in this area drains near the ears and it gets trapped while one sleeps. You'll be back to your old self shortly. Hope this helps and enjoy your new addition.

Yes it is possible. But since you are 1 month post delivery, it is too early to tell. Ideally wait for about 3 months before you can accurately assess.

Congratulations on your new baby!  Thank you for sharing your case and photos.  While it is unlikely that you will have any permanent changes to your eyelids from the swelling, it is important to make sure there isn't any other causes for the condition.  Sometimes, at-home therapies or a short course of medications can help the problem to resolve.  Schedule a consultation with a board certified oculoplastic surgeon (eyelid specialist) to discuss your condition.  Good luck!