The relevant bits of Article 1 Section 3 state that:
Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States
This seems to imply that, if convicted, the Senate may then choose to remove the president, or to bar them from holding future offices. It is not immediately clear that a conviction necessarily implies either of these punishments.
Does impeachment conviction necessarily imply removal from office?