Using a radio frequency tunnel diode oscillator technique, we measured the temperature dependence of the in-plane London penetration depth $Deltalambda_{ab}(T)$ in Fe$_{1+y}$(Te$_{1-x}$Se$_{x})$ 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 $Deltalambda_{ab}(T)$ 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 $napprox2$ and with no systematic dependence on the Se concentration. Most importantly, in the limit of T$rightarrow$0, in some samples we observed a narrow region of linear temperature dependence of penetration depth, suggestive of nodes in the superconducting gap of Fe$_{1+y}$(Te$_{1-x}$Se$_{x})$.
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