After filing a trademark in one country, can I file the same trademark in another?

Yes, it is possible. If you register a trademark in one country and later want to register the same trademark in another country, you can do so by filing another application in the second jurisdiction.

However, when extending the trademark to other countries, it is important to note that the trademark may not be accepted in the territory of the other country. Just because you have successfully registered a trademark in the first country, it does not mean that you will automatically register it successfully somewhere else.

Similarly, if you were to register a trademark in two countries at the same time, it may happen that the trademark will be registered in one country and refused in another.

There is essentially no process for "extending a trademark"; you will simply have to file a new application. Since the applications are filed separately in their respective jurisdictions, the results may vary.

If your first trademark 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 please remember these would still be separate applications.

If you file two identical trademark applications within six months of each other in different countries, it will grant you something called "priority filing". With priority filing, the subsequent trademark application filed in a different country can use the earlier filing date from the prior trademark application. What this would mean, in practice, is that if you file your EU trademark application on the 1st of March and afterwards your US trademark application on the 1st of July, the US trademark application can use the priority filing to gain the filing date of your EU trademark application, which will be the 1st of March instead of 1st of July, effectively granting you earlier protection in the US.