  The existence of smooth but nowhere analytic functions is well-known (du
Bois-Reymond, Math. Ann., 21(1):109-117, 1883). However, smooth solutions to
the heat equation are usually analytic in the space variable. It is also
well-known (Kowalevsky, Crelle, 80:1-32, 1875) that a solution to the heat
equation may not be time-analytic at $t=0$ even if the initial function is real
analytic. Recently, it was shown in \cite{Zha20, DZ20, DP20} that solutions to
the heat equation in the whole space, or half space with zero boundary value,
are analytic in time under essentially optimal conditions. In this paper, we
show that time analyticity is not always true in domains with general boundary
conditions or without suitable growth conditions. More precisely, we construct
two bounded solutions to the heat equation in the half plane which are nowhere
analytic in time. In addition, for any $\delta>0$, we find a solution to the
heat equation on the whole plane, with exponential growth of order $2+\delta$,
which is nowhere analytic in time.
