Any problem solved by a finite automaton is in P
Yes, it is true. In terms of complexity classes,
$$
\text{REG} \subseteq \text{P},
$$
where $\text{REG}$ is the class of regular languages (i.e., problems that can be solved by a finite automaton). More specifically,
$$
\text{REG} \subseteq \text{DTIME}(n), \tag{*}
$$
and $\text{DTIME}(n)$ is a strict subset of $\text{P}$ by the time hierarchy theorem.

The proof of (*) is as follows: for any problem in $\text{REG}$, there is a DFA which solves it. Convert that DFA to a Turing machine with the same states and transition function, which always moves to the right until it sees a blank, and then accepts or rejects. This Turing machine always halts in time exactly $n$.



It's also worth mentioning that $$
\text{REG} = \text{DSPACE}(0) = \text{DSPACE}(k)$$
for any fixed constant $k$.