Nope, it will be wrapped up before the end of episode 14. We won't be wondering about Spock's fate or any such thing between seasons. There will be a cliffhanger leading into the next season, but it will be to set up the next storyline, not to resolve the current one. I think that's pretty clear considering the recent comments from PaleyFest, especially Anthony Rapp's quotes here and here.
I'm not saying that Spock's death will be the cliffhanger. I'm saying that if their plan is to basically reboot the show at the end of the season via time travel - to have Burnham go back in time to when she viewed the supernova as a child and use the "power of a supernova" to make the timeline right (something which was directly hinted at last episode) then killing Spock the episode before would make thematic sense. It would help to cut her off emotionally from her timeline, making her more apt to reset things. It would also trigger an amazing "WTF" freakout in fandom, which would raise the visibility of the show. People would be wondering if they were pressing the reset button or just making Discovery a whole new timeline, and tune in next week to find out.
Basically kill Spock at the end of episode 14, and then have the last episode deal with "fixing" the timeline and Discovery's likely jump into the future.