Python 3, 82 bytes

f=lambda a,*b:[*b]and[min([sum((x-y)**2for x,y in zip(a,q))**.5for q in b]+f(*b))]

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Python 3, 82 bytes

f=lambda a,*b:[*b]and[min([sum((x-y)**2for x,y in zip(a,q))**.5for q in b]+f(*b))]

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Python 3 with SciPy, 58 bytes

lambda*a:min(pdist(a))
from scipy.spatial.distance import*

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