Can confirm that Designer will not allow a second space between two words/characters in any p element as of 06:17 PT.
Edit: Confirmed using macOS 10.12 on Chrome 53 (53.0.2785.116) and Safari 10.0 (12602.1.50.0.10).
Edit 2: Uploaded video below. You’ll hear spacebar keystrokes to demonstrate that repeated presses do not insert extra spaces beyond a single space between two words/characters in headings or text elements inside of the Webflow Designer.