Yes, but the "spike protein" isn't floating around in your blood infecting cells, causing them to mutate. In very simple terms:
The mRNA enters a cell and instructs the cell to make a protein, in this case a protein just like the S, or Spike protein of the virus. This spike protein rises to the surface of the cell. Your immune system recognizes this cell as "infected" and attacks it. Just like dead or attenuated virus vaccines, the spike protein isn't multiplying and spreading, it's just sitting there on a cell, waiting to be destroyed. The body can only make as many spike protein particles as there are mRNA nanoparticles in the vaccine dose.
Nope, spike proteins from the injection given to mice, were found in multiple organ systems 24 hours after injection,
(including blood, brain, heart, inguinal lymph node, kidney, liver, lung, gonads, and spleen)
Source?
Not the National Enquirer, but the British Medical Journal.
https://www.bmj.com/content/373/bmj.n958/rr-1