Version 1
Is there a connected graph containing edges $e_1, e_2, e_3$ such that there is a perfect matching containing any two of the edges but no perfect matching containing all three?
EDIT: Brian M. Scott provides a nice example in his answer.
Version 2
Is there a connected graph in which any two nonincident edges belong to a perfect matching but there are three edges such that no perfect matching contains all three?
EDIT: I provide an example in my answer.