I am trying to create a map template. When I do a 'save as' there is no option for mxt, the only option is mxd.

[BrianHadley]

I am using ArcGIS for Desktop 10.3.1 with a basic license.

[DanPatterson_Retired]

You have this posted in the ArcGIS Pro place... You might want to move it to Managing Data‌

In any event, there is a whole section on map templates 

and from the first line

A map template is a map document (.mxd)

and mxt are existing templates which is fine, as long as it works, just save it where you need it