We obtain a $p$-adic semilinear cell decomposition theorem using methods developed by Denef in [Journal fur die Reine und Angewandte Mathematik, vol. 369 (1986), pp. 154-166]. We also prove that any set definable with quantifiers in $(0, 1, +, =, \lambda_q, \mathbf{P}_n)_{\{n\in\mathbb{N},q\in\mathbb{Q}_p\}}$ may be defined without quantifiers, where $\lambda_q$ is scalar multiplication by $q$ and $\mathbf{P}_n$ is a unary predicate which denotes the nonzero $n$th powers in the $p$-adic field $\mathbb{Q}_p$. Such a set is called a $p$-adic semilinear set in this paper. Some further considerations are discussed in the last section.