Is a discrete set inside a compact space necessarily finite?

It is not true in general. Let $X = \{0\}\cup\{2^{-n}:n\in\mathbb{N}\}$ with the topology inherited from $\mathbb{R}$; then $X$ is compact, and $X\setminus \{0\}$ is an infinite discrete subset of $X$. Of course every closed discrete subset of a compact space is finite, so infinite discrete subsets won’t be closed, but in general they will exist. For instance, the space $X$ just described can be embedded in any infinite compact metric space.

Take the subspace $\{0\}\cup\{\frac1n; n\in\mathbb N\}$ of the real line. It is compact and contains an infinite discrete subspace $\{\frac1n; n\in\mathbb N\}$.

You can construct a similar example as a subspace of unit circle, so this fails for manifolds too. (It suffices to make a quotient of $[0,1]$ by identifying zero and one, which leaves the above subspace unchanged.)

More generally, for arbitrary discrete space you can construct a compactification

Abramodj: I just want to point out that closed discrete subset of a compact space is finite is consequence of a more general fact that a discrete space is compact iff it is finite. Now since closed subset of a compact space is compact, so closed discrete subset of compact space are compact and hence finite.