Is it possible to make a grammar LL($1$) which recognizes palindroms?
No. The language of palindromes over any alphabet with at least two elements is not deterministic, and thus has no LR or LL grammar.
The language of even-length palindromes $L_{epal}=\{ww^R \mid w\in\Sigmạ^*\}$  is a classic example of a non-deterministic context-free language, and you can find the outline of a proof in Hopcroft &amp; Ullman (and other standard texts). The restriction to even-length palindromes simplifies the proof, but it can be extended to the language of all palindromes $L_{pal}=\{w \mid w=w^R, w\in\Sigmạ^* \}$. Without doing more work, though, it's evident that $L_{epal}=L_{pal}\cap (\Sigma\Sigma)^*$, and deterministic CFLs are closed against intersection with regular languages.