It is well known that exponential map in $C^{n\times n}$ will cover all non-sigular matrix $GL(n,C)$, which is a basic fact in Lie group and lie algebra theory, whether it is true for p-adic cases.

Recent I may need facts in classical matrix analysis to be generalized in p-aidc matrix cases. something are really trivial, however some others may have essential difficulties at least to me. it is quite striking fact that entire p-adic map doesn't omit value in $C_{p}$, and the p-adic exponential map can not defined in whole $C_{p}$

Any references and comments will be appreciated.