No, by paragraph 4 of article 50 (citing 2 and 3 as well because 4 refers back to those):
- A Member State which decides to withdraw shall notify the European Council of its intention. In the light of the guidelines provided by the European Council, the Union shall negotiate and conclude an agreement with that State, setting out the arrangements for its withdrawal, taking account of the framework for its future relationship with the Union. That agreement shall be negotiated in accordance with Article 218(3) of the Treaty on the Functioning of the European Union. It shall be concluded on behalf of the Union by the Council, acting by a qualified majority, after obtaining the consent of the European Parliament.
- The Treaties shall cease to apply to the State in question from the date of entry into force of the withdrawal agreement or, failing that, two years after the notification referred to in paragraph 2, unless the European Council, in agreement with the Member State concerned, unanimously decides to extend this period.
- For the purposes of paragraphs 2 and 3, the member of the European Council or of the Council representing the withdrawing Member State shall not participate in the discussions of the European Council or Council or in decisions concerning it.
So specifically, the last part excludes the UK from participating in discussions of extension as well as negotiating and concluding an agreement on behalf of the EU Council.
Or as the author of Article 50, Lord Kerr of Kinlochard, puts it in an interview with Politico (regarding paragraph four):
"It’s very important that Britain is not a third country throughout the whole process of the Article 50 negotiation. Up to the moment when we leave, we are a full member. Therefore you have to have sub-paragraph four, saying that when they’re talking about the divorce, the Brits won’t be in the room and if they are in the room, they won’t be voting."
It depends on your definition of "veto".
Under Article 50(4), the UK does not have a vote in the discussions on whether or not the European Council offers an extension. Thus, in the strictest sense, the UK does not have a veto.
However, under Article 50(3), the extension cannot take place without the UK's consent†. Declining any extension offered can be considered, for all intents and purposes, as effective as a veto.
† The European Union (Withdrawal) (No. 2) Act 2019 limits the power of the UK Prime Minister to decline an extension offered before 31 October 2019.