I start sweating profusely when I'm under a lot of pressure. Is this a sign of a mental illness?

No. Hyperhidrosis (or excess sweating) is more common than you'd think! antiperspirants are the simple answer, but often won't work with true hyperhidrosis. I have had success treating patients underarms, hands, feet, forehead, and scalp with botox. Botox is fda approved for sweating and is very safe and effective. Botox typically lasts for 6 months when used for excess sweating. Hope this helps!
Sweating w stress. Sweating is a component of the neurologic system. When you're stressed, your heart rate may increase, and you may start sweating, and even feel your heart beating in the chest. Your sweating may be normal.
Unlikely. Many people sweat under pressure. Its because your sympathic nervous system is active and this is an expected response.
Coping mechanism. This is a coping mechanism and body reaction due to changesin adrenal hormones. If this is embarrassing you need to develop methods to cope with short tr. Stress such as breathing, meditation or biofeedback.