As an overall approach, I would start at the high level solution architecture first, then progress to the specifics and exact components.
In this case, you need up to a 160 x amplification of a 1v p-p HF (high frequency) RF signal. 8 MHz is radio frequency. 160 v p-p RF brings with it some other challenges around RFI / EMI that need to be considered (you didn’t explain how this 160V RF signal is to be used).
160 x amplification is 40 dB of gain. You will need at least a couple of amplification stages. This sounds similar to “linear amplifiers”, where one would begin with a Class A single transistor amp stage, followed by a Class AB push-pull stage. Then there’s the matter of matching the input and output impedances, properly biasing each stage and ensuring no RF gets back up into your power supply with proper filtering.
At the low current levels you need, it shouldn’t require highly expensive power transistors as you’re only talking 10 watts output. With efficiency in the 60%+ range you can probably do fine with 20 watt finals in the PA.
That’s how I’m framing out a potential solution based on the limited requirements available at this stage.
PCB layout and shielding will also need some consideration, along with your power supply.