"Chan eil Eilean Mhanainn ann am Breatainn."
Translation:The Isle of Man is not in Britain.
Yes, but it is not part of Britain (which is an island) nor is it part of the UK (which is a sovereign territory). This is the first useful fact that I have seen in the sentences in any language so I am right impressed. Let's have more useful facts about Scotland and things relevant to Gaelic (since Manx is very close to Gaelic).
I, and many others, avoid the term British Isles. They are not all British. The Isle of Mann is (as it is a Crown dependency) but part of Ireland isn't (since it was de-colonized in the 1920s). It harks back to a time when the English colonized its neighbours. It is also bad linguistically, as the term British or its variant Brythonic is used to describe the group of languages (of which only Welsh, Cornish and Breton survive, although it was much more widespread once) that was unique to Britain and its emigrés in Brittany (Breatainn Bheag 'Little Britain' as opposed to Great Britain).
"The Isle of Man is not, and never has been, part of the United Kingdom, nor is it part of the European Union. It is not represented at Westminster or in Brussels. The Island is a self-governing British Crown Dependency - as are Jersey and Guernsey in the Channel Islands - with its own parliament, government and laws." https://www.gov.im/about-the-government/departments/cabinet-office/external-relations/constitution/#:~:text=The%20Isle%20of%20Man%20is,own%20parliament%2C%20government%20and%20laws.
True, but I am not quite sure of the relevance as this question is about whether IoM is part of Britain, not whether it is part of the UK. I have already pointed out that they are very different as one is an island and the other is a sovereign territory.
Another issue is the 'and never has been'. This is of course true, but only because of the technicality that the UK is a fairly recent invention. It is certainly not the case that the IoM has always had completely separate government to Britain, as this article on the King of Mann shows. It is simply that there has been no joint rule since the UK was formed. Further, its autonomy has only gradually developed over the centuries, only really reaching its current level in 1949.
Returning to the actual question of the IoM being part of Britain, even that separation has only existed for a while. You may think this is too long to worry about, but since Britain is a geographical concept, not a political one, I think it is quite reasonable to look on a longer timescale. The above link seems to vaguely suggest that the IoM broke off from Britain and Ireland at about the same time. This is not so. Ireland broke off from Europe (including what is now Britain and the IoM) about 16,000 BC, leaving the IoM as part of a European peninsular (that was effectively Britain) from which it separated about 6,500 BC, several hundred years (depending on which source you read) before Britain became an island. This of course can lead to an argument about terminology, but I would suggest it is reasonable to say that the IoM was part of Britain for 9,500 years and has been separate for 8,500 years.
I am not sure of the point you are making. References to Britain are usually geographical and this statement is geographically true. But it is also true politically as the Isle of Mann is an independent country. There has been extensive discussion of the relationship on this page, so please read it, and contribute if you wish.