I will describe the construction of a “square root” anticyclotomic p-adic L-function for symplectic type automorphic representations of the unitary group U(1, 2n-1). This can be seen as a higher dimensional generalisation of the work of Bertolini–Darmon–Prasanna, and one of the main ingredients is the p-adic iteration of Maass–Shimura operators in higher degrees of coherent cohomology. If time permits, I will describe the expected relation with Euler systems outside the region of interpolation.