We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for k-Dominating Set on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connected k-Dominating Set and Total k-Dominating Set (albeit with a worse upper bound on the twin-width). The k-Independent Set problem admits the same lower bound by a much simpler argument, previously observed [ICALP '21], which extends to k-Independent Dominating Set, k-Path, k-Induced Path, k-Induced Matching, etc. On the positive side, we obtain a simple quadratic vertex kernel for Connected k-Vertex Cover and Capacitated k-Vertex Cover on graphs of bounded twin-width. Interestingly the kernel applies to graphs of Vapnik-Chervonenkis density 1, and does not require a witness sequence. We also present a more intricate O(k^1.5) vertex kernel for Connected k-Vertex Cover. Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most optimization/decision graph problems can be solved in polynomial time on graphs of twin-width at most 1.