ThushanDoes the sentence "I have got sick" obviously mean I am still sick

It is not a natural standard English sentence, so I have no idea if it implies that you are still sick or not.

These are OK:
I have been sick.
I got sick.

ThushanDoes the sentence "I have got sick" obviously mean I am still sick ?

In American English we would say, "I have gotten sick". It doesn't say you are still sick, and it doesn't say that you are not still sick. It means that at some time in the past you became sick, i.e., changed from being healthy to being sick. It says nothing about your current state

CalifJimIn American English we would say, "I have gotten sick".

Yes, the "got" as an alternate form of "gotten" is British, or perhaps a regionalism.

In BrE, I believe, the usual phrase to express this idea would be 'I've fallen ill/sick' . And if you're are seriously ill , then you should stick with "ill" (sick indicates a relatively mild condition).

IvanhrIn BrE, I believe, the usual phrase to express this idea would be 'I've fallen ill/sick' . And if you're are seriously ill , then you should stick with "ill" (sick indicates a relatively mild condition).

Yes, that would have to be British. It would not be typical usage in American English.

CJ

In BrE, I believe, the usual phrase to express this idea would be 'I've fallen ill/sick' .

This seems extremely formal to me. I think in everyday British English one would probably say simply
I'm ill/sick. if the condition is true at present,
or