9 Deaths of the Ninja
Jelly, 30 28 26 bytes

»0o⁴+
;@ñ\&lt;1S_9«0N“dead”×?


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How it works

;@ñ\&lt;1S_9«0N“dead”×?  Main link. Input: e (events), h (initial health)

;@                    Prepend h to e.
  ñ\                  Reduce the resulting array by the next, dyadic link.
                      This returns the array of intermediate results.
    &lt;1                Check each intermediate value for non-positivity.
      S               Sum. This calculates the number of event deaths.
       _9             Subtract 9 from the result.
         «0           Take the minimum of the result and 0. This yields 0 if no
                      lives are left, the negated amount of lives otherwise.
                   ?  Conditional:
                  ×     If the product of the minimum and h is non-zero:
           N              Return the negated minimum.
            “dead”      Else, return "dead".


»0o⁴+                 Dyadic helper link. Arguments: x, y

»0                    Take the maximum of x and 0.
                      This yields x if x &gt; 0 and 0 otherwise.
  o⁴                  Take the logical OR of the result and the second input (h).
    +                 Take the sum of the result and y.