Can A type stars have planets with life around them?

It is possible. Earth formed in 1 billion years. However, our planet is big compared to other rocky planets (such as mercury, mars). If it was a smaller planet, say the size of Mercury, and it is located in the "goldilocks zone", then that planet would form and cool down much faster, maybe in 0.7 billion years, leaving 300 million years for life to form. Probably life would stop at the bacterial level. But don't forget that bacteries (not all) can thrive without sunlight. So even though your star had died off, life would continue to exist at the microscopic level.

We will compare them (with Sun) by ratios.

A type star has a typical temperature (surface) of 10000 K. Temperature ratio (with Sun at 5785 K) is

T = 10000 / 5785 = 1.7286 .

L = [RT²]²,

if R is nor relevant (assuming the star to be of same radius as the Sun), or R = 1

& at a distance r from it, (Power) Intensity I would be

I = L/r²

= [T²/r]².

For the same Intensity

T²/r = 1,

r = T²,

In terms of Earth's distance (in AU)

r = 1.7286² = 2.9880 AU ≈ 3 AU.

This is the mean distance of the habitable (Goldilocks) zone.

1 billion years is adequate time for the Life to evolve to a higher form if the planet is Venus or Earth sized. Bigger size would trap more gases & their unwanted compounds that Life could do without & smaller size would denude itself of essential gases in the Atmosphere, even that may be too flimsy to stave off the Star's harmful radiation.

A good mix of O₂ & H₂ will ensure Liquid water & CO₂. If the evolution proceeds steadily & logically without a hitch like what happened on Earth multicellular organisms & even higher forms can be expected in about two hundred million years

You are correct in saying that an A type stars life is much shorter than our suns. In order for life to develope on this planet around the A type star not only would it have to develop much quicker but the planet would also have to be much farther away from the star. This is due to the much greater luminosity of an A type star which is radiating a lot in the ultraviolet. So in short yes, it would be possible although the time required for intelligent life would not be enough so it would be limited to very simple life, maybe even photosynthetic autotrophs? But again the planet would have to be much farther away from the star in order to be in that stars habitable zone.

Would the billion year life be long enough for planets to form, for the primordial planets to cool down enough for liquid water to exist, for the first self replicating molecule to form by random... Probably not.