To be honest, lying is a normal developmental stage that all kids go through, and how you deal with it depends on why they are lying in the first place.
Withmy DD2 at 11, what i have done is simply to state that i know she lied and therefore this will happen (consequence of yuor choosing). Don't bother getting into any kind of debate about it or argueing over whether tehy lied or not etc. If raging about being disappointed isn't working then stop doing it.
With a small child, tbh at that age she probably doesn't fully understand lying. She just wants to get out of something she doesn't want to do. With younger toddlers also they don't understand that they are lying they just want to make mum happy again not cross so they do what they think will work (ie. lie). And there isn't that much to be worried about at age 3 - only start to worry if they get a lot older and keep lying out of habit e.g. an 8 y old lying at least times 4 times a day when it is obvious what they are doing is time to start tackling it seriously.
the only type of lie i get very angry about is a lie aimed to get another innocent person into trouble. otherwise, I don'y get too bothered. Then again, everyones different and lying can really anger and upset some parents, it just isn't a big bother of mine in particular.
With DD2 what worked was saying, 'are you sure that is what happened. Truth tellers get marbles in their jars' (I filled a jar with marbles and they got various treats for a full jar). The incentive to tell the truth offen worked on its own. With DD1 I just didn't ask questions, until a certain stage it was a given that she would lie if caught out because she was utterly terrified of being punished. However i should point out that DD1 and DD2 weren't/aren't ordinary kids and they deal with things differently than others so what worked with htem might not work with other kids at all.
The biggest thing that will work though is not giving the opportunity to lie in the first place. if you know they have done something wrong then don't ask about it, because it gives them the opportunity to lie to you. Just state you did this, so therefore ...