When removing a link, it is important to not just delete the linked text. The link might remain in the HTML-code visible on the frontend, but not in the text editor. Remove it by first marking the linked text in the text editor. Next to the link icon there will now also be a "Remove link" icon (a globe with a broken link). Click this to remove the link. After that remove the text that was linked.
Some text editors does not have the "Remove link" icon. Instead click the "Insert link" icon, which will now have the title text "Modify link". The tab on the left will now be "Remove link". Click to remove the link. After that remove the text that was linked.