Che cosa significa I'll always drop whatever it is.
(feel free to call me any time of day, any time of night) ?

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The person who told you this is telling you that, if you have something important to call them about or tell them about, they will be willing to stop doing whatever they are doing to listen to you and talk to you! This is why they say they will “drop whatever it is,” meaning they will stop what they are doing to talk to you at the time that you call them.
So when they say “feel free to call me any time of day, any time of night,” they are saying that you can call them anytime! So you don’t need to feel restricted or reserved about reaching out to them.