There are some situations when someone ask us to promise to not tell anything about a secret.
Then, if an other person ask me regarding this secret and I react and reply something to not break the promise.
Is this considered as lying?
Or it is permissible ?

Not breaking the promises is a sign of a believer. You may say frankly that you are not supposed to say anything about someone’s secrets instead of saying you don’t anything (because if you know something and say you don’t that would be lie). But it depends even in some situations you can deny specially if it’s related to someone’s life matter or can cause big distractions amongst the families or if you are carrying a sensitive post where your enemy wants to know your secrets so if you simply say at this moment that you know but you won’t tell, you might be tortured or see danger for you life so better to deny at all. But in normal situations amongst families issue, if such situations arise and the other party knows that you know something then you can boldly say yes I know but I can’t say due my religious restrictions rather than making some lame excuses and tell lies make it clear.