That's what would normally happen.
C2 has twice the frequency of C1, therefore a pre-recorded sample at C1 would need to be played twice the speed to attain the pitch of C2. The pitch at C3 is double (twice) the frequency of C2, etc.
This is why multi-samples are used - ie. a fresh, pre-recorded sample used at every octave, or significantly more than one sample per octave for more detailed work, including several sampled velocity layers, etc., particularly for sampling realistic and incredibly expressive instruments like pianos.
Anyhow, use the C1 sample and allow it to stretch from C0-G1 or similar, and set the base note to be C1 as the reference pitch. Then the C2 sample could stretch from G#1-G2, and again set the base note to be C2 in this case; The C3 sample from G#2-G3, with C3 set as the basenote, etc....
Otherwise, you need something with time-stretching if you wish to stretch one particular sample upwards while retaining the time length of the original sample.