By hitting the “Select Features” button (see below), select the desired field for export.
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When the desired area is highlighted in blue, go to “Table of Contents” and right click the right Layer.
Next, search “Data” in this overview and go to “Export Data”.
In the window that now appears you should indicate you only want to export the selected field. If necessary, you can change the name of the new file when saving it. Afterwards, click on “OK”.
You have created a new Layer now. You will keep it until the end of the session. The Layer can be saved as a layer file (under ‘file’ in the menu bar).