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Python 3, 135 129 84 bytes

-4 bytes thanks to Mr. Xcoder!

-42 bytes thanks to Daniel Schepler!

g=lambda s,l:(s in l)or-min(g(w,l)for w in{w[:len(s)+1]for w in l if w[:len(s)]==s})

Try it online!

A 1 indicates the current player will win, whereas a -1 indicates they will lose.