There are companies that have two career streams: technical and management. If you go down technical, you might become a principal engineer, etc. If you go down the managerial, you can pursue a director role. If you feel your company isn't offering that and is unlikely to, you can look for such companies.

Note however, I think a bit of management is inevitable as you become more senior. Being senior means you often have the knowledge and expertise to multiply your team's output through interactions beyond individual contribution. By this I mean, bringing senior expertise to the table when planning, being part of the long term strategy and roadmap discussions, involving the team, seeking out contribution for your projects, etc.