Text-based conversations between users with searchable history in Microsoft Teams Free
In Teams Free there is no option to save or export an individual chat only. However, it is possible to export the entire chat and file history.
https://teams.live.com/dataexport
The exported data will contain information on all chats and files still saved on the Cloud, but it is also possible to view the content of selected chats.
The option to save an individual message is only available in Teams for mobiles. This feature is not available in Teams for desktops or Web.