Can a stroke affect the part of the brain which controls our breathing? If so, can it be fatal? Thanks.

Yes. It depends on the location of stroke . Usually if stroke is in the brain stem( medula ) , can cause respiratory difficulty which could be fatal. Injury to this center may lead to center respiratory failure, which necessitates mechanical ventilation; usually the prognosis is grave.
Could be. Strokes which affect the vertebro-basilar system and cause damage to the brainstem could affect respiratory rhythm rather profoundly, and if massive swelling were a consequence, eventual apneustic pattern could emerge requiring a respirator. Of course, in absence of breathing support, death could ensue.