Using a radio frequency tunnel diode oscillator technique, we measured the temperature dependence of the in-plane London penetration depth ∆λ ab (T ) in Fe1+y(Te1−xSex) single crystals, down to temperatures as low as 50 mK. A significant number of samples, with nominal Se concentration x=0.36, 0.40, 0.43 and 0.45 respectively, were studied and in many cases we found that ∆λ ab (T ) shows an upturn below 0.7 K, indicative of a paramagnetic type contribution. After subtracting the magnetic background, the low temperature behavior of penetration depth is best described by a power law with exponent n ≈ 2 and with no systematic dependence on the Se concentration. Most importantly, in the limit of T→0, in some samples we observed a narrow region of linear temperature dependence of penetration depth, suggestive of nodes in the superconducting gap of Fe1+y(Te1−xSex).