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We examine existence and uniqueness of strong solutions of multi-dimensionalmean-field stochastic differential equations with irregular drift coefficients.Furthermore, we establish Malliavin differentiability of the solution and showregularity properties such as Sobolev differentiability in the initial data aswell as H\"older continuity in time and the initial data. Using the Malliavinand Sobolev differentiability we formulate a Bismut-Elworthy-Li type formulafor mean-field stochastic differential equations, i.e. a probabilisticrepresentation of the first order derivative of an expectation functional withrespect to the initial condition.

Existence and Regularity of Solutions to Multi-Dimensional Mean-Field Stochastic Differential Equations with Irregular Drift