Extreme surface waves in deep-water long-crested sea are often interpreted as a manifestation in real world of the so-called breathing solitons of the focusing nonlinear Schrodinger equation. While the spontaneous emergence of such coherent structures from nonlinear wave dynamics was demonstrated to take place in fiber optics systems, the same point remains far more controversial in the hydrodynamic case. With the aim to shed further light on this matter, the emergence of breather-like coherent wave groups in long-crested random sea is here investigated by means of high-resolution spectral simulations of the fully nonlinear two-dimensional Euler equations. Our study is focused on parametrizing the structure of random wave fields with respect to the Benjamin--Feir index, which is a nondimensional measure of the energy localization in Fourier space. This choice is motivated by previous results, showing that extreme-wave activity in long-crested sea is highly sensitive to this parameter. It is found that coherent wave groups do develop within wave fields characterized by sufficiently narrow-banded spectra, and that such coherent structures closely match realizations of Kuznetsov--Ma breathes in Euler dynamics. The characteristic spatial and temporal scales of wave group dynamics, and the corresponding occurrence of extreme events, are quantified and discussed by mean of space-time autocorrelations of the surface elevation envelope and extreme events statistics.