UK business visa (Standard Visitor Visa) - extension of number of days

You are in the UK on a 'business visa' (assumed to be a Standard Visitor Visa, class "C") and you want to extend your stay...

then, can I apply for an extension from the UK only?

There are no provisions in the rules for extending a visa issued for 6 months (180 days) or less.

Some confusion arises from this paragraph in Appendix V of the rules...

An application for an extension of stay as a visitor must comply with the requirements in paragraphs 34 – 34C of Part 1 of these Rules.

but this is for people whose visas were issued for a limited period, like a month, and the person wants to extend to the full 6 months. You already have the full 6 months. Moreover, it's very subtle, but intrepid followers of this site with an appreciation for irony will point out that Paragraphs 34 - 34C have been deleted. Those rules were quietly removed by Orders in Council two years ago.

then, can I apply for an extension from the UK only?

As mentioned you cannot apply (successfully) for an extension. In your case, the name of the application is "Further Leave to Remain (FLR)". While the possibility is always there to submit any sort of application you want (up to and including British nationality, they have no compunctions whatever about taking the fee and then refusing), applying for FLR as a visitor will invariably cause removal paperwork to be issued and present incredible difficulties in getting another visa. Your biometrics will be annotated (that's a very bad thing).

The controlling technical reference is found in Paragraph 28 of the rules...

28. An applicant for an entry clearance must be outside the United Kingdom and Islands at the time of the application.

Other rules in Part 1 also apply.

Or do I need a fresh visa application from India?

You need to make a fresh application, BUT as long as you are outside of the UK and Islands, it can be in any locale where there is a British mission, including Europe...

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Final note:

Eyes wide to V 4.2 (b)

will not live in the UK for extended periods through frequent or successive visits, or make the UK their main home; and