As of 2020 is there any proof of tiny extra dimensions or observations of them?
I assume that your question is really this: is there any indication that there is a fourth or fifth space dimension, one that may be hidden from easy detection because it is “compact”, that is, curled up.
On the other hand, if you take the word “dimension” to mean the mathematical definition of dimension, then it is trivial that there are numerous other dimensions. Temperature is a dimension; so is color; so is the number of photons. But I don’t think you were referring to this meaning of the word, because if you were, the question (and answer) isn’t very interesting.
A space dimension would be one that could be transformed into one of the existing spatial dimensions through a rotation. It would be a non-reducible dimension in a vector space.
There are some experimental efforts to detect other spatial dimensions, particularly by looking for deviations in gravity at small (meter, cm, mm) dimensions. Nothing has yet given a positive result.
Do other dimensions exist? Many of my theorist friends would vote “yes”. I would vote “no”. Multiple dimensions provide a mechanism for getting rid of some of the infinities of quantum theory, but in my mind, that is a rather weak reason to believe in them. I suspect we’ll find a better way to fix the theory, some day.
I don't think that's a useful reply. If you really wanted to help him by correcting him, then you should at the very least provide the correct word.Proof is for mathematics, not physics.