In the following, $T$ is a bounded operator on a Banach space $X$.

- $T$ is called "power bounded" if $\sup_{n\in \mathbb N}\|T^n\|<\infty$;
- $T$ is called "mean ergodic" if the Cesàro sums $\frac{1}{n}\sum_{k=1}^n T^k$ converge strongly as $n\to \infty$;
- $T$ is called a "Ritt operator" if its spectrum is contained in the unit disk and $\sup_{\{|\lambda|>1\}}\|(\lambda-1)R(\lambda,T)\|<\infty$.

Now, the following is known:

- In a reflexive space every power bounded operator is mean ergodic (Lorch 1939).
- This not true any more if we drop the assumption of reflexivity (counterexample in a large class of non-reflexive Banach spaces by Fonf-Lin-Wojtaszczyk 2001).
- Furthermore, it is known that Ritt property is strictly stronger than power boundedness (Nagy-Zemanek and Lyubich, both 1999).

So in particular in reflexive spaces each Ritt operator is mean ergodic.

Is it known whether Ritt property implies mean ergodicity in general (i.e., possibly non-reflexive) spaces?

I wouldn't bother imposing some geometric conditions on the Banach space, or even restricting to sequence spaces.