  We consider differential delay equations of the form $\partial_tx(t) =
X_{t}(x(t - \tau))$ in $\mathbb{R}^n$, where $(X_t)_{t\in S^1}$ is a
time-dependent family of smooth vector fields on $\mathbb{R}^n$ and $\tau$ is a
delay parameter. If there is a (suitably non-degenerate) periodic solution
$x_0$ of this equation for $\tau=0$, that is without delay, there are good
reasons to expect existence of a family of periodic solutions for all
sufficiently small delays, smoothly parametrized by delay. However, it seems
difficult to prove this using the classical implicit function theorem, since
the equation above is not smooth in the delay parameter. In this paper, we show
how to use the M-polyfold implicit function theorem by Hofer-Wysocki-Zehnder
[HWZ09, HWZ17] to overcome this problem in a natural setup.
