Does the Brouwer fixed point theorem holds for the uncountable power $[0,1]^\kappa$ of the interval, $\kappa>\omega$ ?

That is, does every continuous endomorphism $[0,1]^\kappa\to [0,1]^\kappa$ necessarily have a fixed point ?

A form of the Brouwer fixed point theorem says that for any two maps $f,g:K\to C$ from a compact $K$ to a contractible $C$ necessarily have a coincidence $x\in X$ such that $f(x)=g(x)$ whenever one of them is surjective and has acyclic fibres. Is there a form of this which covers $[0,1]^\kappa$ for $\kappa>\omega$ ? (In (Eilenburg and Montgomery,1946) the precise requirements are: $C$ is an acyclic absolute neighborhood retract, and $K$ is a compact metric space. This does not apply to our case.)

Does the Brouwer fixed point theorem holds for the uncountable power $[0,1]^\kappa$ of the interval, $\kappa\geq\aleph_1$ ?

That is, does every continuous endomorphism $[0,1]^\kappa\to [0,1]^\kappa$ necessarily have a fixed point ?

A form of the Brouwer fixed point theorem says that for any two maps $f,g:K\to C$ from a compact $K$ to a contractible $C$ necessarily have a coincidence $x\in X$ such that $f(x)=g(x)$ whenever one of them is surjective and has acyclic fibres. Is there a form of this which covers $[0,1]^\kappa$ for $\kappa\geq\aleph_1$ ? (In (Eilenburg and Montgomery,1946) the precise requirements are: $C$ is an acyclic absolute neighborhood retract, and $K$ is a compact metric space. This does not apply to our case.)