It is known that under some topological conditions, algebraic algebras over a field are necessarily algebraic of bounded degree. A simple proof of this, valid for power-associative algebras is given by \textit{B. Cuartero} and \textit{J. E. Galé} [Commun. Algebra 22, 329-337 (1994; Zbl 0837.17002)]. This proof, however, does not go over to the Lie algebra setting. Here, a subset \(V\) of a Lie algebra \(L\) over a field \(K\) is said to be weakly algebraic if for any \(x \in L\), \(y \in V\), \(x\) is annihilated by some polynomial (depending on \(x\) and \(y)\) in ad \(y\). \(L\) itself is called algebraic if for any \(y \in L\), there is a polynomial annihilating ad \(y\). The authors prove that if \(L\) is a Baire Lie algebra over a complete non- discrete valuated field \(K\) and if \(V\) is a non-empty open subset of \(L\) which is weakly algebraic, then \(L\) is algebraic of bounded degree. A similar result is proved also for restricted Lie algebras and \(p\)- polynomials. Moreover, the technique used allows the authors to obtain general results on algebraicity of bounded degree for arbitrary nonassociative algebras (with a convenient definition of algebraic elements).