I think all kids go through this stage, even when it is made very clear to them from the off that lying is unacceptable. Dd2 in particular has done this (and sometimes still does). Things that work for us:
If she owns up and admits what she did wrong, the first thing we say is, "thank you for telling the truth. That was a good thing to do. I am very unhappy that you jumped off the bed, because [insert reason]. But I am pleased that you didn't lie to me about it".
If she lies about what she has done, and it's fairly obvious when she does, we point out that we know she is lying, and tell her why. "I know you are lying because your eyes look different/your body language changes when you lie" - that way she knows that "I know you're lying" isn't just a stab in the dark. (However she can't tell just what gives her away so she can't cover up!!)
If she's done something naughty and lied about it, then we make it clear that whatever punishment she receives, it is for the lie, not the naughty deed. So far this has never tripped us up 
If, like dd2, your dd is very emotional, you can appeal to her emotions. Recently, I left a pile of clothes on dd2's sofa to be put away, and later noticed that they had gone. I asked her if she had put them away and she said yes, so I praised her for doing it without being asked. Later, I found the clothes thrown into a cupboard. Rather than read her the riot act, I told her that she had really upset me, because it made me feel that she must have been laughing at me when I praised her for putting them away, because she knew she hadn't. That really got to her and she was really upset that she had upset me.
Hope some of this is useful!