No — there is no reliable public source currently confirming that Val Kilmer is left-handed.

Val Edward Kilmer is an American actor. He began his career on the stage and rose to prominence in the mid-1980s with roles in comedy films like 'Top Secret!' (1984) and 'Real Genius' (1985). He went on to star in blockbusters such as 'Top Gun' (1986), 'Willow' (1988), and 'The Doors' (1991), where he portrayed Jim Morrison.

Kilmer's diverse filmography also includes the role of Batman in 'Batman Forever' (1995), 'Heat' (1995), and 'The Saint' (1997). He is known for his intense method acting and versatility across various genres. In later years, he continued to work in film and television, often taking on supporting roles.

In 2015, Kilmer was diagnosed with throat cancer, which severely impacted his ability to speak. He underwent a tracheostomy and has been open about his recovery journey, chronicling it in his memoir, 'I'm Your Huckleberry' (2020), and the documentary 'Val' (2021).

Despite health challenges, Kilmer remains an active figure in the arts, continuing to express himself through painting, writing, and various creative projects. His resilience and contributions to cinema have earned him a dedicated following.