Extending Johannes' answer a little further: Let $A$ be a separable unital $C^*$-algebra and denote by $\mathbb{K}$ the compact operators on a separable infinite dimensional Hilbert space. Let $M(A \otimes \mathbb{K})$ be the multiplier algebra of the stabilization $A \otimes \mathbb{K}$ of $A$.

By a theorem of Mingo the unitary group $U(M(A \otimes \mathbb{K}))$ of this multiplier algebra is contractible when equipped with the norm topology. In this topology it is also a Banach Lie group with Lie algebra the real Banach space of skew-adjoint operators in $M(A \otimes \mathbb{K})$. Let $e_1$ be a rank 1 projection in $\mathbb{K}$ and let $p = 1 \otimes e_1$. Then we have $$ p\; M(A \otimes \mathbb{K})\; p \cong p\;(A \otimes \mathbb{K})\;p \cong A $$ and $U(pM(A \otimes \mathbb{K})p)$ is a Banach Lie subgroup of $U(M(A \otimes \mathbb{K}))$. Using the exponential map (which exists and is not that bad for Banach Lie groups), we get that $U(M(A \otimes \mathbb{K})) / U(pM(A \otimes \mathbb{K})p)$ allows a local section. Therefore $$ U(M(A \otimes \mathbb{K})) \to U(M(A \otimes \mathbb{K})) / U(pM(A \otimes \mathbb{K})p) $$ is a model for $EG \to BG$ for $G = U(A)$, the unitary group of the $C^*$-algebra.