Given $ f \left( x \right) = \left\| x \right\| $ is a norm function its Prox is given by (For any Norm):
$$ \operatorname{Prox}_{\lambda f \left( \cdot \right)} \left( v \right) = v - \lambda \operatorname{Proj}_{ \mathcal{B}_{ \left\| \cdot \right\|_{\ast} } } \left( \frac{v}{\lambda} \right) $$
Where $ \operatorname{Proj}_{ \mathcal{B}_{ \left\| \cdot \right\|_{\ast} } } \left( \cdot \right) $ is the Orthogonal Projection Operator and $ \mathcal{B}_{ \left\| \cdot \right\|_{\ast} } $ is the Norm Unit Ball (Of the Dual Norm).
In your case we're dealing with the $ {L}_{2} $ Norm which is self dual.
Moreover, the Projection onto the $ {L}_{2} $ Unit Ball is given by:
$$ \operatorname{Proj}_{ \mathcal{B}_{ \left\| \cdot \right\|_{2} } } \left( x \right) = \begin{cases}
\frac{x}{ \left\| x \right\|_{2} } & \text{ if } \left\| x \right\|_{2} > 1 \\
x & \text{ if } \left\| x \right\|_{2} \leq 1
\end{cases} $$
In summary:
$$ \operatorname{Prox}_{\lambda \left\| \cdot \right\|_{2}} \left( v \right) = v - \lambda \operatorname{Proj}_{ \mathcal{B}_{ \left\| \cdot \right\|_{2} } } \left( \frac{v}{\lambda} \right) = \begin{cases}
v - \lambda \frac{ \frac{v}{\lambda} }{ \left\| \frac{v}{\lambda} \right\|_{2} } & \text{ if } \left\| \frac{v}{\lambda} \right\|_{2} > 1 \\
v - \lambda \frac{v}{\lambda} & \text{ if } \left\| \frac{v}{\lambda} \right\|_{2} \leq 1
\end{cases} = \left(1 - \frac{\lambda}{ \left\| v \right\|_{2} } \right)_{+} v $$
Where $ \left( x \right)_{+} = \max \left\{ 0, x \right\} $.