Yes, it is possible. If you register a trade mark in one country and later want to register the same trade mark in another country, you can do so by filing another application in the second jurisdiction.
However, when extending the trade mark to other countries, it is important to note that the trade mark may not be accepted in the territory of the other country. Just because you have successfully registered a trade mark in the first country does not mean you will automatically register it successfully somewhere else.
Similarly, if you were to register a trade mark in two countries at the same time, it may happen that the trade mark will be registered in one country and refused in another.
There is essentially no process for “extending a trade mark”; you will simply have to file a new application. Since the applications are filed separately in their respective jurisdictions, the results may vary. It is important to undertake searches in all countries you wish for protection and to register your trade mark before applying to do so.
If your first trade mark was registered in a WIPO member state, you can use the Madrid System to file your second application in another member state. This can come in handy if you want to expand to more countries at once, but remember these are still separate applications.
If you file two identical trade mark applications within six months of each other in different countries, it will grant you something called “priority filing.” With priority filing, the subsequent trade mark application filed in a different country can use the earlier filing date from the prior trade mark application. For example, if you file your EU trade mark application on the 4th April and afterward your US trade mark application on the 4th August, the US trade mark application can use the priority filing to gain the filing date of your EU trade mark application, which will be the 4th April instead of the 4th August. This means the subsequent application can use the filing date of the first application, providing earlier protection in the new country.