Hi there, I'm writing a letter in formal style and I'm looking for a word/phrase means "a little bit" that fit in a formal context. Any recommendation?
Depending on the context, it might be possible to replace ' a little bit' with 'some' (which, arguably, should also be avoided in formal contexts, whenever possible).

Still, I see no reason why a little bit' can't be used in a formal register.

If I knew the specific context of your letter, perhaps I'd be able to come up with something better ( rather than just offering you this, vague and imprecise answer)..
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I am about to write "I am a little bit disappointed when I heard the result", but it sound informal, so I want to phrase it more formally.
It sounds fine to me, but you should write 'I was a little bit...'.

You could say, " I was somewhat disappointed ...." I hope that helps.

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It is possible that somewhat is a good substitute for a little bit.