The first search for nonresonant production of Higgs boson pairs (HH) with one H decaying into four leptons and the other into a pair of b quarks is presented, using proton-proton collisions recorded at a center-of-mass energy of s = 13 TeV by the CMS experiment. The analyzed data correspond to an integrated luminosity of 138 fb−1. A 95% confidence level upper limit of 32.4 is set on the signal strength modifier μ, defined as the ratio of the observed HH production rate in the HH→ ZZ∗b b ¯ → 4 ℓb b ¯ decay channel to the standard model (SM) expectation. Possible modifications of the H trilinear coupling λ HHH with respect to the SM value are investigated. The coupling modifier κλ, defined as λ HHH divided by its SM prediction, is constrained to be within the observed (expected) range −8.8 (−9.8) < κλ < 13.4 (15.0) at 95% confidence level. [Figure not available: see fulltext.].