  We derive a stochastic Gronwall lemma with suprema over the paths in the
upper bound of the assumed affine-linear growth assumption. This allows
applications to It\^o processes with coefficients which depend on earlier time
points such as stochastic delay equations or Euler-type approximations of
stochastic differential equations. We apply our stochastic Gronwall lemma with
path-suprema to stochastic functional differential equations and prove a strong
convergence rate for coefficient functions which depend on path-suprema.
