If a function is injective and continuously differentiable, then is it a bijection? If this isn't the case then I'd appreciate a counterexample, although for the time being a simple yes or no would suffice.

No. For instance, the function $f:(0,1)\to \mathbb{R}$ given by $f(x)=x$ is injective and continuously differentiable but not surjective.

A bit less trivially, you can even have a counterexample $\mathbb{R}\to\mathbb{R}$, such as $f(x)=e^x$.