What is the ruling on lying to escape showing off, such as when a fasting person says, "I am not fasting" to escape showing off?

It is not permissible to lie to avoid showing off. However, one may use ma'arid (allusive speech or dissimulation) when there is a need or a legitimate benefit. For instance, there is no harm in a person saying "Praise be to Allah" to someone who invited him to eat, while intending that he is fasting. However, it is not considered showing off to inform an inquirer about one's fasting if asked, as it has been narrated in the Sunnah for a fasting person to say: "Indeed, I am fasting." It is also permissible to announce one's fasting with a good intention, such as encouraging others to follow suit.