This questions is about the distinction between:

• Lie groupoids: we require source and target maps to be submersions. This implies that the domain of the composition map, $G_1 \;{}_s\!\times_t G_1$, is a smooth manifold.

• Groupoids internal to the category of smooth manifolds. Here we only require that the fibre product $G_1 \;{}_s\!\times_t G_1$ exists. Recall that transversality is sufficient but not necessary.

I was trying to construct - explicitly - a groupoid internal to smooth manifolds that is not a Lie groupoid, but failed so far. Can somebody provide one?

For example, I tried to set $G_0 := \mathbb{R}$ and $G_1 := \mathbb{R} \times \{0,1\}$, where the morphisms of type $\{0\}$ are identity morphisms, and the morphisms of type $\{1\}$ are endomorphisms of the object $0$. The fibre product $G_1 \;{}_s\!\times_t G_1$ exists (it is $\mathbb{R} \times \{0,1\}^2$), and the composition is smooth and associative. So this is a category internal to smooth manifolds whose source and target maps are not submersive. However, the morphisms of type $\{1\}$ are not invertible, because the identity morphisms of the object $0$ is of type $\{0\}$.