GeoExt3 is dual licensed. You will normally use it under GPLv3. If you own a Sencha ExtJS Commercial License, you have the right to use GeoExt3 under the BSD License.
Details: https://www.sencha.com/legal/#Sencha_Ext_JS
See the Sencha official link for details: https://www.sencha.com/legal/open-source-faq/
GeoExt3 is GPLv3, but you have the right to use it and extend it with a Sencha ExtJS Commercial License under BSD.
The license you choose depends on the license of the framework ExtJS you use with it. If you are creating open source applications and you are using the GPL v3 version of the Ext JS framework, then the appropriate GeoExt license is the GPL v3 license for GeoExt. If you are creating closed source applications and you are using the commercial version of the Ext JS framework, then the appropriate GeoExt license is the BSD license for GeoExt, which can be combined with the commercial version of Ext JS for a closed source application.