Find distance between the closest 3D points

R, 34 bytes

function(...)min(dist(rbind(...)))

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This is a nice opportunity to use R's ... syntax to define a function that can accept a variable number of arguments; in this case, the x,y,z coordinates of each point.

The dist function calculates the pairwise distance between all rows of a matrix, using a chosen method - luckily, the default is 'euclidean' and so isn't specified in this case.

Of course, it could be even shorter if we allow the input to already be combined-together as a matrix, but this wouldn't be so neat...

R, 23 bytes

function(m)min(dist(m))

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Wolfram Language (Mathematica), 33 bytes

Min[Norm[#-#2]&@@@#~Subsets~{2}]&

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Python 3, ^{106 95} 93 bytes

Saved 11 bytes thanks to fireflame241!!!
Saved 2 bytes thanks to Jonathan Allan!!!

lambda l:min(sum((a-b)**2for a,b in zip(l[p],v))**.5for q,v in enumerate(l)for p in range(q))

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Inputs a list of points as tuples and returns the Euclidean distance between the two closest points.
Works with points of any dimension so long as they are consistent within the list.