The following may be a bug or me not understanding p-adic floating point 
computations:


sage: R = Qp(2,type='floating-point',print_mode='terse')
sage: M = Matrix(R,4,[0, 0, 1, 1, 2^20, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1])
sage: M.det()
1048575
sage: M.inverse()
---------------------------------------------------------------------------
ZeroDivisionError                         Traceback (most recent call last)
....

ZeroDivisionError: input matrix must be nonsingular

sage: M = Matrix(R,4,[0, 0, 1, 1, 2^19, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1])
sage: M.inverse()
[     1048575            2            0            1]
[274878955520       524288            1      1048575]
[      524289       524287            1       524287]
[274877382656       524289      1048575       524289]

Works. 

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