Sports Climbing Competition is always stressful. All competitions are. We prepare for years hoping for that perfect 3-4 minutes during competitions. Those 3-4 minutes are actually dependent on Mental Strength.
Climbing that hard project is also stressful. The process of trying and failing on an outdoor climb until you climb it is called Redpointing. It is a process that plays on a mind game. It may take years to climb a route and sometimes just a handful of attempts.
In short, Mental Strength is really important for climbing!
From my experience, Mental Strength is not a gifted skill. One has to achieve that by practice and practice. Include Mental Training sessions along with Physical Training sessions.
So, start by creating your Mental Training Plan!