Marta here, from Trello support.
I'm afraid it's currently not possible to automatically archive cards, sorry.
What I usually do is hover on top of the card and press "c", which will immediately archive the card.
That being said, I’ll pass your interest in this on to our team for consideration—let me know if there's any additional info you'd like for me to send their way.
If you're curious about how our team decides on what requests we work on, this article gives some insight into that process: http://help.trello.com/article/724-submitting-feature-requests-for-trello
If you don't mind using a 3rd-party (paid) tool, Butler for Trello can automate the archival of cards after a number of days in a list.
This help article shows how to set this up: https://help.butlerfortrello.com/article/48-archive-trello-cards-automatically-after-a-number-of-days
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