We present the first example of non‐classical hydrogen bonding in Cu^I complexes. To this end, we have studied the coordination capacity of the tripodal phosphine ligands HTIM(PR₂)₃ {tris[1‐(di‐R‐phosphanyl)‐3‐methyl‐1H‐indol‐2‐yl]methane, R = Ph, iPr} towards copper, leading to a series of eight Cu^I complexes [{HTIM(PR₂)₃}CuX] [R = Ph (1–4), iPr (5–8); X = Cl^–, I^–, OTf^–, BF₄^–]. An anagostic interaction in the Cu^I complexes has been proposed based on the results of ¹H NMR, ultrahigh‐resolution X‐ray diffraction, and theoretical [atoms‐in‐molecules (AIM) and natural bonding orbital (NBO)] analyses of the electron density. Among the different synthesized complexes, [{HTIM(PPh₂)₃}CuOTf] (2) shows the strongest Cu···H–C interaction with a linear geometry (179.98°) and the shortest distance described so far [d(Cu–H) = 1.9 Å] in its toluene solvated form. Both parameters are characteristic of 3c–4e^– interactions. In addition, we have found that these tripodal ligands exhibit unprecedented selectivity towards Cu^I ions. The synthesized HTIM(PR₂)₃Cu^I complexes have been found to be catalytically active and selective for the hydroboration of CO₂ to formic acid.