The question is exactly that in the title: Does every infinite $\Sigma^1_1$ set of natural numbers have an infinite $\Delta^1_1$ subset?
Some background: The lower-level analog of this question, Does every infinite c.e. set have an infinite computable subset?, is easily seen to be true. However, this is in some sense the wrong analog of the question I want to ask. The correct intuition in higher recursion theory tends to be that $\Delta^1_1$ corresponds to finite, as opposed to computable, and that the correct analog of "c.e." is $\Pi^1_1$ rather than $\Sigma^1_1$. This makes me guess that the answer to my question is "no," since the only reason I have for favoring "yes" is based on an incorrect analogy.
I also imagine that the answer (whether a counterexample or a proof) is fairly simple. This is why I have posted on MathSE as opposed to MathOF. I have looked at Sacks' book "Higher Recursion Theory" (which can be gotten freely at Project Euclid: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pl/1235422631), but I have not been able to find the answer there.
This question has no broad significance; I am just interested in understanding the analytic hierarchy better, and this seemed like a problem I ought to understand.
Thank you very much for your help!