We prove that the continuous cohomology groups of a $p$-adic reductive group with coefficients in an admissible unitary $\mathbb{Q}_p$-Banach space representation $\Pi$ are finite-dimensional and compare them to certain Ext-groups. As an application of this result, we show that the continuous cohomology of $SL_2(\mathbb{Q}_p) $ with values in non-ordinary irreducible $\mathbb{Q}_p$-Banach space representations of $GL_2(\mathbb{Q}_p)$ vanishes.