I see a little generalization of idea proposed by Robert Israel.
Let $\mathcal{A}$ be unital $C^*$-algebra such that there exist two elements $s_1$, $s_2$ with property $s_k^*s_l=\delta_k^l 1_{\mathcal{A}}$ (this is anlogues of isometries with pairwise orthgonal images). Consider elements $S_n=s_1^{n-1}s_2$. It is easy to check that $ S_k^*S_l=\delta_k^l 1_{\mathcal{A}} \tag{1} $ i.e. we built a family of isometries with pairwise orthogonal images.
Let $Y$ be unital left $\mathcal{A}$-module and $y\in Y\setminus\{0\}$. Consider elements $\{S_n\cdot y:n\in\mathbb{N}\}$. Assume that $\sum_{k=1}^N \lambda_k S_k\cdot y=0$. Multiplying this equality by $S_n^*$ where $n\in\{1,\ldots,N\}$ with usage of (1) we obtain $\lambda_n y=0$. Since $y\neq 0$, then $\lambda_n=0$ for all $n\in\{1,\ldots,N\}$. This means that $\{S_n\cdot y:n\in\mathbb{N}\}$ linearly independent and $\operatorname{dim}Y=+\infty$.