Yes, this is true. The simple reason is that bounded functions form a normal family. Therefore their derivatives are uniformly bounded on every compact.
To obtain the estimate $|f'(z)|<1$, apply Cauchy theorem: $$|f'(z)|=\left|\frac{1}{2\pi}\int_{|\zeta-z|=1}\frac{f(\zeta) d\zeta}{(\zeta-z)^2}\right|\leq 1.$$ Equality in the right inequality can happen only if $f$ is constant, so we always have a strict inequality.