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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