Multi-patch spline parametrizations are used in geometric design and isogeometric analysis to represent complex domains. Typically, quadrilateral patches are adopted in both frameworks. We consider the particular class of multi-patch parametrizations that are analysis-suitable G1 (AS-G1), which is a specific geometric continuity definition which allows to construct, on the multi-patch domain, C1 isogeometric spaces with optimal approximation properties (cf. Collin et al., 2016). It was demonstrated in Kapl et al. (2018) that AS-G1 multi-patch parametrizations are suitable for modeling complex planar multi-patch domains. We construct a local basis, and an associated dual basis, for a specific C1 isogeometric spline space A over a given AS-G1 multi-patch parametrization. The space A is C1 across interfaces and C2 at all vertices, and is therefore a subspace of the entire C1 isogeometric space V1. At the same time, A allows optimal approximation of traces and normal derivatives along the interfaces and reproduces all derivatives up to second order at the vertices. In contrast to V1, the dimension of A does not depend on the domain parametrization. This paper also contains numerical experiments which exhibit the optimal approximation order in L2 and L∞ of the isogeometric space A and demonstrate the applicability of our approach for isogeometric analysis. •We define a C1-smooth isogeometric subspace over multi-patch domains.•The isogeometric subspace is C1 across edges and C2 at vertices.•We present a basis construction for the isogeometric subspace.•The isogeometric subspace displays optimal approximation properties.