Does there exist an infinite noncommutative division ring with finite center?
Yes. The following construction is attributed to Hilbert in "A first course in noncommutative rings" by T.Y. Lam., p. 217. Let $k$ be a field with an automorphism $\sigma$. Write $D = k((x,\sigma))$ for the noncommutative ring of formal Laurent series $\sum_{i = n}^\infty a_ix^i$ with twisted multiplication rule $xa = \sigma(a) x$ for $a\in k$. Then $D$ is a division ring. If $k_0$ is the fixed field of $\sigma$, then either $Z(D) = k_0$ or $Z(D) = k_0((x^s))$ depending on whether $\sigma$ has infinite order or finite order $s$, respectively. Thus all we need to do is choose $\sigma$ and $k$ such that $k_0$ is a finite field and $\sigma$ has infinite order. This can be done by choosing say the Frobenius automorphism of $\overline{\mathbb{F}}_p$. 

