In QGIS 3.14 I used the "identify features tool" or (Ctrl + Shift + i) then select the country. The Identify Results box on the right should appear with all the details of the selected object. Right-click inside this box and select Copy Feature. Go to the main Edit Dropdown menu and Paste Feature As ... New layer. The layer can then be saved in format and CRS you require by selecting the newly created layer ... right-click and Export - Save Feature As. Note to select the country you may need to make the layer with the countries editable using the toggle editing button (with a pencil in it)
You can use one of the following processing tools:
- Extract selected features (recommended if you want to select it visually, as you did)
- Extract by Attribute (recommended if you know the countries Name)
- Extract by Location (recommended if you only know the location and you have a second layer representing this location (point, line or polygon))
- Extract by expression (recommended if you need to perform an expression, e.g. choose by several conditions)
You can find all these tools by searching the processing toolbox. If you cannot find the toolbox, go to "View" --> "Panels" and check "Processing Toolbox", it should then appear on the right of your QGIS window.