The 21-year-old right-back's previous deal at The Den expired after Tuesday night's defeat at Barnsley.
Smith initially joined the Lions in late October and has made 18 appearances so far.
Boss Kenny Jackett told the club website: "We're delighted Adam is staying.
"He's been a regular at right-back for us and has done very well. We look forward to continuing to work with him and help with his development and football education here."
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