Is it permissible to smile at a non-mahram person when speaking with them?

It is forbidden for a woman to smile in the face of a non-mahram man if there is a fear of fitnah (temptation). She should not pay attention to people thinking she is arrogant, because keeping non-mahrams away from her is what is required by Islamic law. Dealing politely prevents seducing those with sick hearts, as in the Almighty’s saying: "Then do not be soft in speech [to men], lest he in whose heart is disease should covet, but speak with appropriate speech," and His saying, the Almighty: "They should draw over themselves their outer garments. That is more suitable that they will be known and not be abused." And if a non-mahram man thinks she is arrogant, this is a bad assumption on his part, and he may be held accountable for it. Not smiling does not mean frowning, but rather adhering to proper conduct and avoiding fitnah to preserve society.