Do we have evidence of heritable personality traits in chimpanzees?
There has been some research in order to explore this direction, for instance, see this reference:

A. Weiss et al. - "The Heritability of Personality Factors in Chimpanzees", Behavior Genetics, Vol. 30, No. 3, 2000., 

in which the authors study five human-like factors - Surgency,
Dependability, Emotional Stability, Agreeableness, and Openness. 

Quoting verbatim their findings:


  Only Dominance ... was significantly heritable. Dependability showed evidence that it might be heritable, but the parameter estimates were not statistically significant ... Two other personality traits, Agreeableness and Emotional Stability, showed positive heritability estimates but the parameter
  estimates were nonsignificant and small. Openness and Surgency showed no evidence of heritability. 


In fact, the statistical correlation between dominance and heritability makes a strong sense.  Here's what I make out of this - this was a correlation based study, or kind of an experiment, hence the stress on statistical significance. But the question it raises is - is there a causal relationship between the two, i.e. does there exist any proper mechanism for the inheritance of "dominance". Finding such a mechanism was clearly beyond the scope of this study, but I believe the requisite answer for your question is if one stumbles on to such mechanisms, and not statistical correlations, which are mere pointers. 

But on the basis of the study at least, all that one can safely conclude is - statistical correlations suggest that inheritance of "dominance" appears plausible, but the exact mechanism of how this implements out in practice, is beyond the scope of this study.