Thank  you  all!  It  would take too much bandwidth to mention all the
people who have really helped me solve the problem.

My initial peak detector circuit looked like this:

            +----------------------+
            |                      |
            |       |+5V           |
            |  +---------+         |          +----
            +--| _       |         |          |
               |    OP262|         |          |
 ---510k--o----| +       |--|>|----o------0---|  PIC
          |    |         |         |      |   |
         10k   +---------+ 1N4148  |      |   |
          |         |             ---    1M   +----
          |        ===            ---     |
         ===               22 nF   |      |
                                  ===    ===

The main problem was in the input stage of amp. When input signal was
much lower then capacitor voltage, which is the sitution most of the
time,  the  capacitor  was  shunted  by  saturated input stage and 10k
resistor. That's why it looked as if capacitor had 15k in parallel.
That  was  my  mistake  assuming high input impedance of opamp for all
cases. Thanks Gerry and Wagner for pointing me in that direction.

I would like to keep the circuit simple, without any trannies or
additional amps. OP262 is good enough for driving caps because it
doesn't limit output current (but bad thing if you shorten output :)
And I don't need long hold-up time, just about 100 us, so that PIC ADC
could aquire the signal (I know the time when expect the peak).

So now the circuit changed this way:
                +---|>|---+ 1N4148
           D2   |         |
            +---o--620k---o--------+
            |                      |
            |       |+5V           |
            |  +---------+         |          +----
            +--| _       |  D1     |          |
               |    OP262|         |          |
 ---510k--o----| +       |--|>|----o----------|  PIC
          |    |         |         |          |
         10k   +---------+ 1N4148  |          |
          |         |             ---         +----
          |        ===            ---
         ===               10 nF   |
                                  ===
I decided to throw away the 1M resistor. It has no purpose if
capacitor has a path to discharge (620k and 10k). Without the diode D1
in the feedback, the circuit loses ability to catch small pulses. D1
helps a lot. When input signal is getting over the cap voltage, the
amp input stage is saturated and this makes the amp drive the cap
hard. This circuit now works as expected. Effects of diode leakage,
cap's internal resistance, ADC impedance, etc don't have significant
impact on performance. I could even decrease the cap value in half.

I will check all the suggested parts.

Thanks again,
 Nikolai